problem solving adding and subtracting polynomials

Adding and Subtracting Polynomials

A polynomial looks like this:

To add polynomials we simply add any like terms together ... so what is a like term?

Like Terms are terms whose variables (and their exponents such as the 2 in x 2 ) are the same.

In other words, terms that are "like" each other.

Note: the coefficients (the numbers you multiply by, such as "5" in 5x) can be different.

are all like terms because the variables are all x

are all like terms because the variables are all xy 2

Example: These are NOT like terms because the variables and/or their exponents are different:

Adding polynomials.

• Place like terms together
• Add the like terms

Example: Add   2x 2 + 6x + 5   and   3x 2 - 2x - 1

Here is an animated example:

(Note: there was no "like term" for the -7 in the other polynomial, so we didn't have to add anything to it. )

Adding in Columns

We can also add them in columns like this:

Adding Several Polynomials

We can add several polynomials together like that.

Example: Add    (2x 2 + 6y + 3xy)  ,   (3x 2 - 5xy - x)   and   (6xy + 5)

Line them up in columns and add:

Using columns helps us to match the correct terms together in a complicated sum.

Subtracting Polynomials

To subtract Polynomials, first reverse the sign of each term we are subtracting (in other words turn "+" into "-", and "-" into "+"), then add as usual.

Note: After subtracting 2xy from 2xy we ended up with 0, so there is no need to mention the "xy" term any more.

• Inspiration

Addition and Subtraction of Polynomials

Adding and subtracting polynomials.

The basic component of a polynomial is a monomial . When we add or subtract polynomials, we are actually dealing with the addition and subtraction of individual monomials that are similar or alike.

What is a Monomial?

A monomial can be a single number, a single variable, or the product of a number and one or more variables that contain whole number exponents .

This means that the exponents are neither negative nor fractional.

• [latex]7[/latex]
• [latex]y[/latex]
• [latex]2x[/latex]
• [latex] – \,9xy[/latex]
• [latex] – \,4x{y^2}[/latex]
• [latex]10{x^2}{y^3}{z^4}[/latex]
• [latex]{3 \over 4}{k^5}{m^2}h{r^{12}}[/latex]

So now we are ready to define what a polynomial is.

What is a Polynomial?

A polynomial can be a single monomial or a combination of two or more monomials connected by the operations of addition and subtraction .

A polynomial has “special” names depending on the number of monomials or terms in the expression. More so, the degree of a polynomial with a single variable is determined by the largest whole number exponent among the variables.

For example:

Examples of How to Add and Subtract Polynomials

Example 1 : Simplify by adding the polynomial expressions

The key in both adding and subtracting polynomials is to make sure that each polynomial is arranged in standard form . It means that the powers of the variables are in decreasing order from left to right .

Observe that each polynomial in this example is already in standard form, so we no longer need to perform that preliminary step.

Now, there are two ways we can proceed from here.

• First, we can add this the “usual” way, that is, add them horizontally .

I suggest that you first group similar terms in parenthesis before performing addition.

• Another way of simplifying this is to add them vertically .

Place the  similar terms in the same column before performing addition.

As you can see, the answers in both methods came out to be the same!

Example 2 : Simplify by adding the polynomial expressions

Notice that the first polynomial is already in the standard form because the exponents are in decreasing order. However, the second polynomial is not! We must first rearrange the powers of [latex]x[/latex] in decreasing order from left to right.

• Then add them horizontally…

Similar or like terms are placed in the same parenthesis.

• Or add them vertically…

Similar or like terms are placed in the same column before performing the addition operation.

Example 3 : Simplify by adding the polynomial expressions

We are given two trinomials to add. But first, we have to “fix” each one of them by expressing it in standard form.

Add only similar terms. Notice that the [latex]y[/latex] -variables with exponents [latex]3[/latex] and [latex]2[/latex] are placed in their own parenthesis to avoid the accidental addition of non-similar terms.

• Adding Polynomials – Horizontally

Let’s check our work if the answer comes out the same when we add them vertically.

• Adding Polynomials – Vertically

Example 4 : Simplify by adding the polynomial expressions

Let’s add the polynomials above vertically . Align like terms in the same column then proceed with polynomial addition as usual.

Example 5 : Simplify by subtracting the polynomial expressions

Subtracting polynomials is as easy as changing the operation to normal addition. However, always remember to also switch the signs of the polynomial being subtracted.

This is how it looks when we rewrite the original problem from subtraction to addition with some changes on the signs of each term of the second polynomial.

Original Problem

Rewritten Problem

• The original subtraction operation is replaced by addition .
• The second polynomial is “tweaked” by reversing the original sign of each term.

At this point, we can proceed with our normal addition of polynomials. Make sure that similar terms are grouped together inside a parenthesis.

Subtracting Polynomials – Horizontally

Subtracting Polynomials – Vertically

We can also subtract the polynomials in a vertical way. First, convert the original subtraction problem into its addition problem counterpart as shown by the green arrow . Make sure to align similar terms in a column before performing addition.

Example 6 : Simplify by subtracting the polynomial expressions

The two polynomials that we are about to subtract are not in standard form. Begin by rearranging the powers of variable [latex]x[/latex] in decreasing order. Change the operation from subtraction to addition, align similar terms, and simplify to get the final answer.

Transform each polynomial in standard form

Subtract by switching the signs of the second polynomial, and then add them together.

Example 7 : Simplify by subtracting the polynomial expressions

In this problem, we are going to perform the subtraction operation twice.

That means we also need to flip the signs of the two polynomials which are the second and third.

Perform regular addition using columns of similar or like terms.

Example 8 : Simplify by adding and subtracting the polynomials

Rewrite each polynomial in the standard format. Replace subtraction with addition while reversing the signs of the polynomial in question. Finally, organize like or similar terms in the same column and proceed with regular addition.

Adding the polynomials vertically, we have…

Example 9 : Simplify by adding and subtracting the polynomials

If we add the polynomials vertically, we have…

You might also like these tutorials:

• Dividing Polynomials using Long Division Method
• Dividing Polynomials using Synthetic Division Method
• Multiplying Binomials using FOIL Method
• Multiplying Polynomials

Adding and Subtracting Polynomials

While addition and subtraction of polynomials, we simply add or subtract the terms of the same power. The power of variables in a polynomial is always a whole number, power can not be negative, irrational, or a fraction. It is straightforward to add or subtract two polynomials. A  polynomial  is a mathematics expression written in the form of \(a_0x^n + a_1x^{n-1} + a_2x^{n-2} + ...... + a_nx^{0}\).

The above expression is also called polynomial in  standard form , where \(a_0, a_1, a_2.........a_n\) are constants, and n is a  whole number . For example x 2  + 2x + 3,   5x 4  - 4x 2  + 3x +1 and 7x - √3 are polynomials.

How Can We Add Polynomials?

The addition of polynomials is simple. While adding polynomials, we simply add like terms. We can use columns to match the correct terms together in a complicated sum. Keep two rules in mind while performing the addition of polynomials.

• Rule 1:   Always take like terms together while performing addition .
• Rule 2:   Signs of all the polynomials remain the same.

For example, Add 2x 2 + 3x +2 and 3x 2 - 5x -1

• Step 1: Arranging the polynomial in standard form. In this case, they are already in their standard forms.
• Step 2:  Like terms in the above two polynomials are: 2x 2  and 3x 2 ; 3x and -5x; 2 and -1.
• Step 3: Calculations with signs remaining same:

Like Terms are terms whose variables, along with their exponents, are the same. For example, 2x, 7x, -2x, etc are all like variables.

Unlike Terms

Unlike Terms are terms whose either variables, exponents, or both variables and exponents are the not same. For example, 2, 7x 2 , -2y 2 , etc are all unlike variables.

Subtraction of Polynomials

The subtraction of polynomials is as simple as the addition of polynomials. Using columns would help us to match the correct terms together in a complicated subtraction . While subtracting polynomials, separate the like terms and simply subtract them. Keep two rules in mind while performing the subtraction of polynomials.

• Rule 1:  Always take like terms together while performing subtraction.
• Rule 2:  Signs of all the terms of the subtracting polynomial will change, + changes to - and - changes to +.

For example, we have to subtract 2x 2 + 3x +2 from 3x 2 - 5x -1

• Step 2:  Like terms in the above two polynomials are: 2x 2  and 3x 2 ;3x and -5x;2 and  -1
• Step 3:  Enclose the part of the polynomial which to be deducted in parentheses with a negative (-) sign prefixed. Then, remove the parentheses by changing the sign of each term of the polynomial expression.
• Step 4: Calculations after altering the signs of the subtracting polynomials:

Steps for Adding and Subtracting Polynomials

The addition or subtraction of polynomials is very simple to perform, all we need to do is to keep some steps in mind. To perform the addition and subtraction operation on the polynomials, the polynomials can be arranged vertically for complex expressions. For simpler calculations, we can perform the operation using the horizontal arrangement.

Adding and Subtracting Polynomials Horizontally

Polynomials can be added and subtracted in horizontal arrangement using the steps given below,

• Step 1: Arrange the polynomials in their standard form.
• Step 2: Place the polynomial next to each other horizontally. 
• Step 3: First separate the like terms. 
• Step 4:  Arrange the like terms together.
• Step 5:  Signs of all the polynomials remain the same in addition. While in Subtraction, the signs of the terms in subtracting polynomial change.
• Step 6:  Perform the calculations.

Adding and Subtracting Polynomials Vertically

Polynomials can be added and subtracted in vertical arrangement using the steps given below,

• Step 1: Arrange the polynomials in their standard form
• Step 2: Place the polynomials in a vertical arrangement, with the like terms placed one above the other in both the polynomials.
• Step 3: We can represent the missing power term in the standard form with "0" as the coefficient to avoid confusion while arranging terms.
• Step 4:  Signs of all the polynomials remain the same in addition. While in Subtraction, the signs of the terms in subtracting polynomial change.
• Step 5:  Perform the calculations

By following these steps we can solve adding and subtracting polynomials.

Example: (3x 3 + x 2 - 2x -1) + (x 3  + 6x + 3).

The given polynomials are arranged in their standard forms.

Addition performed horizontally:

• Step 1:  Separate the like terms: 3x 3  and x 3 ; x 2 ; -2x and 6x; -1 and 3
• Step 2:  Arrange the like terms together: 3x 3  + x 3 + x 2  + (-2x + 6x) + (-1 + 3)
• Step 3: Perform the calculations: (3 + 1)x 3  + x 2  + (-2 + 6)x + (-1 + 3)= 4x 3  + x 2  + 4x + 2

Addition performed vertically:

• Step 1:  Arrange both the polynomials one above the other with like terms place one above the other. We can represent the missing power term in the standard form with "0" as the coefficient to avoid confusion while arranging terms.
• Step 2: Perform the calculations.

\[ \begin{align} \ \ 3x^3 + x^2 - 2x -1 \\ + \ x^3 + 0x^2 + 6x + 3 \\ \hline \\ 4x^3 + x^2 + 4x + 2 \\ \hline \end{align}\]

• The highest power of the variable in a polynomial is called the degree of the polynomial. 
• The algebraic expressions having negative or irrational power of the variable are not polynomials.
• Addition and subtraction in polynomials can only be performable on like terms. 

Challenging Question on Adding and Subtracting Polynomials

Solved Examples

Example 1: Add two polynomials, 3x + 2y, and 4y + 5z to find the solution. 

Given polynomials are (3x + 2y) + (4y + 5z). Here like terms are only 2y and 4y. So addition can only be performed on these two terms, the other terms 3x and 5z will not get affected. 3x + (2y + 4y) + 5z = 3x + (2 + 4)y + 5z = 3x + 6y + 5z

Therefore, answer is 3x + 6y + 5z.

Example 2: Add the polynomials 3x 2 + 4y 2  - 2z 2  + 1 and -x 2  -7y 2  + 3, and subtract the result from 5x 2  + y 2 - 8z 2  - 6, to find if the sum of coefficients of all the variables is 9.

Let us add the first two polynomials.

(3x 2 + 4y 2  - 2z 2  + 1) + (-x 2  -7y 2  + 3)

= (3 - 1)x 2  + (4 - 7)y 2  - 2z 2  + (1 + 3)

= 2x 2  - 3y 2  -2z 2  + 4

Subtract the above polynomial

5x 2  + y 2  - 8z 2 - 6 - (2x 2  - 3y 2  -2z 2  + 4)

= 5x 2  + y 2  - 8z 2 - 6 - 2x 2  + 3y 2  + 2z 2  - 4

= (5 - 2)x 2  + (1 + 3)y 2  + (-8 + 2)z 2  - 10

= 3x 2  + 4y 2  - 6z 2  - 10

Sum of the coefficients of all the variables is 3 + 4 - 6 = 1. Therefore, No, the sum of all coefficients of variables is not 9.

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FAQs on Adding and Subtracting Polynomials

How do we add or subtract polynomials.

Adding or subtracting polynomials is simple. While adding or subtracting polynomials we need to keep the rules for adding and subtracting a polynomial in mind. The rules can be explained as,

• Rule 1:   Always take like terms together while performing addition or subtraction.
• Rule 2: Signs of all the polynomials remain the same in addition. While in Subtraction, the signs of the subtracting polynomials change.

What are Binomials?

Binomials are polynomials that contain only two terms. For example x 2  + y 2  and 3x + 2y are binomials. For example, x + y +  z is not a binomial.

What is the Main Thing to Remember When you are Adding and Subtracting Polynomials?

The main thing to remember while performing addition and subtraction on polynomials is:

• to keep in mind the concept of like terms
• when a polynomial multiplied with a negative sign, all the signs will be changed. i.e., + to - and - to +

How do you Combine Like Terms?

While  combining like terms , such as 2x and 7x, we simply add their coefficients. For example, 2x + 7x = (2+7)x = 9x.

What are Like Terms?

Can you combine terms with different exponents.

No, you can only combine terms with the exact same variable and the exact same exponent. That means you can only combine squared variable terms with squared variable terms, cubed variable terms with cubed variable terms, etc.

6.1 Add and Subtract Polynomials

Learning objectives.

By the end of this section, you will be able to:

• Identify polynomials, monomials, binomials, and trinomials
• Determine the degree of polynomials
• Add and subtract monomials
• Add and subtract polynomials
• Evaluate a polynomial for a given value

Be Prepared 6.1

Before you get started, take this readiness quiz.

Simplify: 8 x + 3 x . 8 x + 3 x . If you missed this problem, review Example 1.24 .

Be Prepared 6.2

Subtract: ( 5 n + 8 ) − ( 2 n − 1 ) . ( 5 n + 8 ) − ( 2 n − 1 ) . If you missed this problem, review Example 1.139 .

Be Prepared 6.3

Write in expanded form: a 5 . a 5 . If you missed this problem, review Example 1.14 .

Identify Polynomials, Monomials, Binomials and Trinomials

You have learned that a term is a constant or the product of a constant and one or more variables. When it is of the form a x m a x m , where a a is a constant and m m is a whole number, it is called a monomial. Some examples of monomial are 8 , −2 x 2 , 4 y 3 , and 11 z 7 8 , −2 x 2 , 4 y 3 , and 11 z 7 .

A monomial is a term of the form a x m a x m , where a a is a constant and m m is a positive whole number.

A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.

Polynomials

polynomial —A monomial, or two or more monomials combined by addition or subtraction, is a polynomial.

• monomial —A polynomial with exactly one term is called a monomial.
• binomial —A polynomial with exactly two terms is called a binomial.
• trinomial —A polynomial with exactly three terms is called a trinomial.

Here are some examples of polynomials.

Polynomial b + 1 4 y 2 − 7 y + 2 4 x 4 + x 3 + 8 x 2 − 9 x + 1 Monomial 14 8 y 2 −9 x 3 y 5 −13 Binomial a + 7 4 b − 5 y 2 − 16 3 x 3 − 9 x 2 Trinomial x 2 − 7 x + 12 9 y 2 + 2 y − 8 6 m 4 − m 3 + 8 m z 4 + 3 z 2 − 1 Polynomial b + 1 4 y 2 − 7 y + 2 4 x 4 + x 3 + 8 x 2 − 9 x + 1 Monomial 14 8 y 2 −9 x 3 y 5 −13 Binomial a + 7 4 b − 5 y 2 − 16 3 x 3 − 9 x 2 Trinomial x 2 − 7 x + 12 9 y 2 + 2 y − 8 6 m 4 − m 3 + 8 m z 4 + 3 z 2 − 1

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial , binomial , and trinomial when referring to these special polynomials and just call all the rest polynomials .

Example 6.1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.

• ⓐ 4 y 2 − 8 y − 6 4 y 2 − 8 y − 6
• ⓑ −5 a 4 b 2 −5 a 4 b 2
• ⓒ 2 x 5 − 5 x 3 − 9 x 2 + 3 x + 4 2 x 5 − 5 x 3 − 9 x 2 + 3 x + 4
• ⓓ 13 − 5 m 3 13 − 5 m 3

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

ⓐ 5 b 5 b ⓑ 8 y 3 − 7 y 2 − y − 3 8 y 3 − 7 y 2 − y − 3 ⓒ −3 x 2 − 5 x + 9 −3 x 2 − 5 x + 9 ⓓ 81 − 4 a 2 81 − 4 a 2 ⓔ −5 x 6 −5 x 6

ⓐ 27 z 3 − 8 27 z 3 − 8 ⓑ 12 m 3 − 5 m 2 − 2 m 12 m 3 − 5 m 2 − 2 m ⓒ 5 6 5 6 ⓓ 8 x 4 − 7 x 2 − 6 x − 5 8 x 4 − 7 x 2 − 6 x − 5 ⓔ − n 4 − n 4

Determine the Degree of Polynomials

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0—it has no variable.

Degree of a Polynomial

The degree of a term is the sum of the exponents of its variables.

The degree of a constant is 0.

The degree of a polynomial is the highest degree of all its terms.

Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.

Example 6.2

Find the degree of the following polynomials.

• ⓐ 10 y 10 y
• ⓑ 4 x 3 − 7 x + 5 4 x 3 − 7 x + 5
• ⓓ −8 b 2 + 9 b − 2 −8 b 2 + 9 b − 2
• ⓔ 8 x y 2 + 2 y 8 x y 2 + 2 y

Find the degree of the following polynomials:

ⓐ −15 b −15 b ⓑ 10 z 4 + 4 z 2 − 5 10 z 4 + 4 z 2 − 5 ⓒ 12 c 5 d 4 + 9 c 3 d 9 − 7 12 c 5 d 4 + 9 c 3 d 9 − 7 ⓓ 3 x 2 y − 4 x 3 x 2 y − 4 x ⓔ −9 −9

ⓐ 52 52 ⓑ a 4 b − 17 a 4 a 4 b − 17 a 4 ⓒ 5 x + 6 y + 2 z 5 x + 6 y + 2 z ⓓ 3 x 2 − 5 x + 7 3 x 2 − 5 x + 7 ⓔ − a 3 − a 3

Add and Subtract Monomials

You have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficient.

Example 6.3

Add: 25 y 2 + 15 y 2 25 y 2 + 15 y 2 .

Add: 12 q 2 + 9 q 2 . 12 q 2 + 9 q 2 .

Add: −15 c 2 + 8 c 2 . −15 c 2 + 8 c 2 .

Example 6.4

Subtract: 16 p − ( −7 p ) 16 p − ( −7 p ) .

Subtract: 8 m − ( −5 m ) . 8 m − ( −5 m ) .

Subtract: −15 z 3 − ( −5 z 3 ) . −15 z 3 − ( −5 z 3 ) .

Remember that like terms must have the same variables with the same exponents.

Example 6.5

Simplify: c 2 + 7 d 2 − 6 c 2 c 2 + 7 d 2 − 6 c 2 .

Add: 8 y 2 + 3 z 2 − 3 y 2 . 8 y 2 + 3 z 2 − 3 y 2 .

Try It 6.10

Add: 3 m 2 + n 2 − 7 m 2 . 3 m 2 + n 2 − 7 m 2 .

Example 6.6

Simplify: u 2 v + 5 u 2 − 3 v 2 u 2 v + 5 u 2 − 3 v 2 .

Try It 6.11

Simplify: m 2 n 2 − 8 m 2 + 4 n 2 . m 2 n 2 − 8 m 2 + 4 n 2 .

Try It 6.12

Simplify: p q 2 − 6 p − 5 q 2 . p q 2 − 6 p − 5 q 2 .

Add and Subtract Polynomials

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

Example 6.7

Find the sum: ( 5 y 2 − 3 y + 15 ) + ( 3 y 2 − 4 y − 11 ) . ( 5 y 2 − 3 y + 15 ) + ( 3 y 2 − 4 y − 11 ) .

Try It 6.13

Find the sum: ( 7 x 2 − 4 x + 5 ) + ( x 2 − 7 x + 3 ) . ( 7 x 2 − 4 x + 5 ) + ( x 2 − 7 x + 3 ) .

Try It 6.14

Find the sum: ( 14 y 2 + 6 y − 4 ) + ( 3 y 2 + 8 y + 5 ) . ( 14 y 2 + 6 y − 4 ) + ( 3 y 2 + 8 y + 5 ) .

Example 6.8

Find the difference: ( 9 w 2 − 7 w + 5 ) − ( 2 w 2 − 4 ) . ( 9 w 2 − 7 w + 5 ) − ( 2 w 2 − 4 ) .

Try It 6.15

Find the difference: ( 8 x 2 + 3 x − 19 ) − ( 7 x 2 − 14 ) . ( 8 x 2 + 3 x − 19 ) − ( 7 x 2 − 14 ) .

Try It 6.16

Find the difference: ( 9 b 2 − 5 b − 4 ) − ( 3 b 2 − 5 b − 7 ) . ( 9 b 2 − 5 b − 4 ) − ( 3 b 2 − 5 b − 7 ) .

Example 6.9

Subtract: ( c 2 − 4 c + 7 ) ( c 2 − 4 c + 7 ) from ( 7 c 2 − 5 c + 3 ) ( 7 c 2 − 5 c + 3 ) .

Try It 6.17

Subtract: ( 5 z 2 − 6 z − 2 ) ( 5 z 2 − 6 z − 2 ) from ( 7 z 2 + 6 z − 4 ) ( 7 z 2 + 6 z − 4 ) .

Try It 6.18

Subtract: ( x 2 − 5 x − 8 ) ( x 2 − 5 x − 8 ) from ( 6 x 2 + 9 x − 1 ) ( 6 x 2 + 9 x − 1 ) .

Example 6.10

Find the sum: ( u 2 − 6 u v + 5 v 2 ) + ( 3 u 2 + 2 u v ) ( u 2 − 6 u v + 5 v 2 ) + ( 3 u 2 + 2 u v ) .

Try It 6.19

Find the sum: ( 3 x 2 − 4 x y + 5 y 2 ) + ( 2 x 2 − x y ) ( 3 x 2 − 4 x y + 5 y 2 ) + ( 2 x 2 − x y ) .

Try It 6.20

Find the sum: ( 2 x 2 − 3 x y − 2 y 2 ) + ( 5 x 2 − 3 x y ) ( 2 x 2 − 3 x y − 2 y 2 ) + ( 5 x 2 − 3 x y ) .

Example 6.11

Find the difference: ( p 2 + q 2 ) − ( p 2 + 10 p q − 2 q 2 ) ( p 2 + q 2 ) − ( p 2 + 10 p q − 2 q 2 ) .

Try It 6.21

Find the difference: ( a 2 + b 2 ) − ( a 2 + 5 a b − 6 b 2 ) ( a 2 + b 2 ) − ( a 2 + 5 a b − 6 b 2 ) .

Try It 6.22

Find the difference: ( m 2 + n 2 ) − ( m 2 − 7 m n − 3 n 2 ) ( m 2 + n 2 ) − ( m 2 − 7 m n − 3 n 2 ) .

Example 6.12

Simplify: ( a 3 − a 2 b ) − ( a b 2 + b 3 ) + ( a 2 b + a b 2 ) ( a 3 − a 2 b ) − ( a b 2 + b 3 ) + ( a 2 b + a b 2 ) .

Try It 6.23

Simplify: ( x 3 − x 2 y ) − ( x y 2 + y 3 ) + ( x 2 y + x y 2 ) ( x 3 − x 2 y ) − ( x y 2 + y 3 ) + ( x 2 y + x y 2 ) .

Try It 6.24

Simplify: ( p 3 − p 2 q ) + ( p q 2 + q 3 ) − ( p 2 q + p q 2 ) ( p 3 − p 2 q ) + ( p q 2 + q 3 ) − ( p 2 q + p q 2 ) .

Evaluate a Polynomial for a Given Value

We have already learned how to evaluate expressions. Since polynomials are expressions, we’ll follow the same procedures to evaluate a polynomial . We will substitute the given value for the variable and then simplify using the order of operations.

Example 6.13

Evaluate 5 x 2 − 8 x + 4 5 x 2 − 8 x + 4 when

• ⓐ x = 4 x = 4
• ⓑ x = −2 x = −2
• ⓒ x = 0 x = 0

Try It 6.25

Evaluate: 3 x 2 + 2 x − 15 3 x 2 + 2 x − 15 when

• ⓐ x = 3 x = 3
• ⓑ x = −5 x = −5

Try It 6.26

Evaluate: 5 z 2 − z − 4 5 z 2 − z − 4 when

• ⓐ z = −2 z = −2
• ⓑ z = 0 z = 0
• ⓒ z = 2 z = 2

Example 6.14

The polynomial −16 t 2 + 250 −16 t 2 + 250 gives the height of a ball t t seconds after it is dropped from a 250 foot tall building. Find the height after t = 2 t = 2 seconds.

Try It 6.27

The polynomial −16 t 2 + 250 −16 t 2 + 250 gives the height of a ball t t seconds after it is dropped from a 250-foot tall building. Find the height after t = 0 t = 0 seconds.

Try It 6.28

The polynomial −16 t 2 + 250 −16 t 2 + 250 gives the height of a ball t t seconds after it is dropped from a 250-foot tall building. Find the height after t = 3 t = 3 seconds.

Example 6.15

The polynomial 6 x 2 + 15 x y 6 x 2 + 15 x y gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with x = 4 x = 4 feet and y = 6 y = 6 feet.

Try It 6.29

The polynomial 6 x 2 + 15 x y 6 x 2 + 15 x y gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with x = 6 x = 6 feet and y = 4 y = 4 feet.

Try It 6.30

The polynomial 6 x 2 + 15 x y 6 x 2 + 15 x y gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with x = 5 x = 5 feet and y = 8 y = 8 feet.

Access these online resources for additional instruction and practice with adding and subtracting polynomials.

• Add and Subtract Polynomials 1
• Add and Subtract Polynomials 2
• Add and Subtract Polynomial 3
• Add and Subtract Polynomial 4

Section 6.1 Exercises

Practice makes perfect.

Identify Polynomials, Monomials, Binomials, and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

ⓐ 81 b 5 − 24 b 3 + 1 81 b 5 − 24 b 3 + 1 ⓑ 5 c 3 + 11 c 2 − c − 8 5 c 3 + 11 c 2 − c − 8 ⓒ 14 15 y + 1 7 14 15 y + 1 7 ⓓ 5 ⓔ 4 y + 17 4 y + 17

ⓐ x 2 − y 2 x 2 − y 2 ⓑ −13 c 4 −13 c 4 ⓒ x 2 + 5 x − 7 x 2 + 5 x − 7 ⓓ x 2 y 2 − 2 x y + 8 x 2 y 2 − 2 x y + 8 ⓔ 19

ⓐ 8 − 3 x 8 − 3 x ⓑ z 2 − 5 z − 6 z 2 − 5 z − 6 ⓒ y 3 − 8 y 2 + 2 y − 16 y 3 − 8 y 2 + 2 y − 16 ⓓ 81 b 5 − 24 b 3 + 1 81 b 5 − 24 b 3 + 1 ⓔ −18 −18

ⓐ 11 y 2 11 y 2 ⓑ −73 −73 ⓒ 6 x 2 − 3 x y + 4 x − 2 y + y 2 6 x 2 − 3 x y + 4 x − 2 y + y 2 ⓓ 4 y + 17 4 y + 17 ⓔ 5 c 3 + 11 c 2 − c − 8 5 c 3 + 11 c 2 − c − 8

In the following exercises, determine the degree of each polynomial.

ⓐ 6 a 2 + 12 a + 14 6 a 2 + 12 a + 14 ⓑ 18 x y 2 z 18 x y 2 z ⓒ 5 x + 2 5 x + 2 ⓓ y 3 − 8 y 2 + 2 y − 16 y 3 − 8 y 2 + 2 y − 16 ⓔ −24 −24

ⓐ 9 y 3 − 10 y 2 + 2 y − 6 9 y 3 − 10 y 2 + 2 y − 6 ⓑ −12 p 4 −12 p 4 ⓒ a 2 + 9 a + 18 a 2 + 9 a + 18 ⓓ 20 x 2 y 2 − 10 a 2 b 2 + 30 20 x 2 y 2 − 10 a 2 b 2 + 30 ⓔ 17

ⓐ 14 − 29 x 14 − 29 x ⓑ z 2 − 5 z − 6 z 2 − 5 z − 6 ⓒ y 3 − 8 y 2 + 2 y − 16 y 3 − 8 y 2 + 2 y − 16 ⓓ 23 a b 2 − 14 23 a b 2 − 14 ⓔ −3 −3

ⓐ 62 y 2 62 y 2 ⓑ 15 ⓒ 6 x 2 − 3 x y + 4 x − 2 y + y 2 6 x 2 − 3 x y + 4 x − 2 y + y 2 ⓓ 10 − 9 x 10 − 9 x ⓔ m 4 + 4 m 3 + 6 m 2 + 4 m + 1 m 4 + 4 m 3 + 6 m 2 + 4 m + 1

In the following exercises, add or subtract the monomials.

7x 2 + 5 x 2 7x 2 + 5 x 2

4y 3 + 6 y 3 4y 3 + 6 y 3

−12 w + 18 w −12 w + 18 w

−3 m + 9 m −3 m + 9 m

4a − 9 a 4a − 9 a

− y − 5 y − y − 5 y

28 x − ( −12 x ) 28 x − ( −12 x )

13 z − ( −4 z ) 13 z − ( −4 z )

−5 b − 17 b −5 b − 17 b

−10 x − 35 x −10 x − 35 x

12 a + 5 b − 22 a 12 a + 5 b − 22 a

14x − 3 y − 13 x 14x − 3 y − 13 x

2 a 2 + b 2 − 6 a 2 2 a 2 + b 2 − 6 a 2

5 u 2 + 4 v 2 − 6 u 2 5 u 2 + 4 v 2 − 6 u 2

x y 2 − 5 x − 5 y 2 x y 2 − 5 x − 5 y 2

p q 2 − 4 p − 3 q 2 p q 2 − 4 p − 3 q 2

a 2 b − 4 a − 5 a b 2 a 2 b − 4 a − 5 a b 2

x 2 y − 3 x + 7 x y 2 x 2 y − 3 x + 7 x y 2

12a + 8 b 12a + 8 b

19y + 5 z 19y + 5 z

Add: 4 a , −3 b , −8 a 4 a , −3 b , −8 a

Add: 4x , 3 y , −3 x 4x , 3 y , −3 x

Subtract 5 x 6 from − 12 x 6 5 x 6 from − 12 x 6 .

Subtract 2 p 4 from − 7 p 4 2 p 4 from − 7 p 4 .

In the following exercises, add or subtract the polynomials.

( 5 y 2 + 12 y + 4 ) + ( 6 y 2 − 8 y + 7 ) ( 5 y 2 + 12 y + 4 ) + ( 6 y 2 − 8 y + 7 )

( 4 y 2 + 10 y + 3 ) + ( 8 y 2 − 6 y + 5 ) ( 4 y 2 + 10 y + 3 ) + ( 8 y 2 − 6 y + 5 )

( x 2 + 6 x + 8 ) + ( −4 x 2 + 11 x − 9 ) ( x 2 + 6 x + 8 ) + ( −4 x 2 + 11 x − 9 )

( y 2 + 9 y + 4 ) + ( −2 y 2 − 5 y − 1 ) ( y 2 + 9 y + 4 ) + ( −2 y 2 − 5 y − 1 )

( 8 x 2 − 5 x + 2 ) + ( 3 x 2 + 3 ) ( 8 x 2 − 5 x + 2 ) + ( 3 x 2 + 3 )

( 7 x 2 − 9 x + 2 ) + ( 6 x 2 − 4 ) ( 7 x 2 − 9 x + 2 ) + ( 6 x 2 − 4 )

( 5 a 2 + 8 ) + ( a 2 − 4 a − 9 ) ( 5 a 2 + 8 ) + ( a 2 − 4 a − 9 )

( p 2 − 6 p − 18 ) + ( 2 p 2 + 11 ) ( p 2 − 6 p − 18 ) + ( 2 p 2 + 11 )

( 4 m 2 − 6 m − 3 ) − ( 2 m 2 + m − 7 ) ( 4 m 2 − 6 m − 3 ) − ( 2 m 2 + m − 7 )

( 3 b 2 − 4 b + 1 ) − ( 5 b 2 − b − 2 ) ( 3 b 2 − 4 b + 1 ) − ( 5 b 2 − b − 2 )

( a 2 + 8 a + 5 ) − ( a 2 − 3 a + 2 ) ( a 2 + 8 a + 5 ) − ( a 2 − 3 a + 2 )

( b 2 − 7 b + 5 ) − ( b 2 − 2 b + 9 ) ( b 2 − 7 b + 5 ) − ( b 2 − 2 b + 9 )

( 12 s 2 − 15 s ) − ( s − 9 ) ( 12 s 2 − 15 s ) − ( s − 9 )

( 10 r 2 − 20 r ) − ( r − 8 ) ( 10 r 2 − 20 r ) − ( r − 8 )

Subtract ( 9 x 2 + 2 ) ( 9 x 2 + 2 ) from ( 12 x 2 − x + 6 ) ( 12 x 2 − x + 6 ) .

Subtract ( 5 y 2 − y + 12 ) ( 5 y 2 − y + 12 ) from ( 10 y 2 − 8 y − 20 ) ( 10 y 2 − 8 y − 20 ) .

Subtract ( 7 w 2 − 4 w + 2 ) ( 7 w 2 − 4 w + 2 ) from ( 8 w 2 − w + 6 ) ( 8 w 2 − w + 6 ) .

Subtract ( 5 x 2 − x + 12 ) ( 5 x 2 − x + 12 ) from ( 9 x 2 − 6 x − 20 ) ( 9 x 2 − 6 x − 20 ) .

Find the sum of ( 2 p 3 − 8 ) ( 2 p 3 − 8 ) and ( p 2 + 9 p + 18 ) ( p 2 + 9 p + 18 ) .

Find the sum of ( q 2 + 4 q + 13 ) ( q 2 + 4 q + 13 ) and ( 7 q 3 − 3 ) ( 7 q 3 − 3 ) .

Find the sum of ( 8 a 3 − 8 a ) ( 8 a 3 − 8 a ) and ( a 2 + 6 a + 12 ) ( a 2 + 6 a + 12 ) .

Find the sum of ( b 2 + 5 b + 13 ) ( b 2 + 5 b + 13 ) and ( 4 b 3 − 6 ) ( 4 b 3 − 6 ) .

Find the difference of ( w 2 + w − 42 ) ( w 2 + w − 42 ) and ( w 2 − 10 w + 24 ) ( w 2 − 10 w + 24 ) .

Find the difference of ( z 2 − 3 z − 18 ) ( z 2 − 3 z − 18 ) and ( z 2 + 5 z − 20 ) ( z 2 + 5 z − 20 ) .

Find the difference of ( c 2 + 4 c − 33 ) ( c 2 + 4 c − 33 ) and ( c 2 − 8 c + 12 ) ( c 2 − 8 c + 12 ) .

Find the difference of ( t 2 − 5 t − 15 ) ( t 2 − 5 t − 15 ) and ( t 2 + 4 t − 17 ) ( t 2 + 4 t − 17 ) .

( 7 x 2 − 2 x y + 6 y 2 ) + ( 3 x 2 − 5 x y ) ( 7 x 2 − 2 x y + 6 y 2 ) + ( 3 x 2 − 5 x y )

( −5 x 2 − 4 x y − 3 y 2 ) + ( 2 x 2 − 7 x y ) ( −5 x 2 − 4 x y − 3 y 2 ) + ( 2 x 2 − 7 x y )

( 7 m 2 + m n − 8 n 2 ) + ( 3 m 2 + 2 m n ) ( 7 m 2 + m n − 8 n 2 ) + ( 3 m 2 + 2 m n )

( 2 r 2 − 3 r s − 2 s 2 ) + ( 5 r 2 − 3 r s ) ( 2 r 2 − 3 r s − 2 s 2 ) + ( 5 r 2 − 3 r s )

( a 2 − b 2 ) − ( a 2 + 3 a b − 4 b 2 ) ( a 2 − b 2 ) − ( a 2 + 3 a b − 4 b 2 )

( m 2 + 2 n 2 ) − ( m 2 − 8 m n − n 2 ) ( m 2 + 2 n 2 ) − ( m 2 − 8 m n − n 2 )

( u 2 − v 2 ) − ( u 2 − 4 u v − 3 v 2 ) ( u 2 − v 2 ) − ( u 2 − 4 u v − 3 v 2 )

( j 2 − k 2 ) − ( j 2 − 8 j k − 5 k 2 ) ( j 2 − k 2 ) − ( j 2 − 8 j k − 5 k 2 )

( p 3 − 3 p 2 q ) + ( 2 p q 2 + 4 q 3 ) − ( 3 p 2 q + p q 2 ) ( p 3 − 3 p 2 q ) + ( 2 p q 2 + 4 q 3 ) − ( 3 p 2 q + p q 2 )

( a 3 − 2 a 2 b ) + ( a b 2 + b 3 ) − ( 3 a 2 b + 4 a b 2 ) ( a 3 − 2 a 2 b ) + ( a b 2 + b 3 ) − ( 3 a 2 b + 4 a b 2 )

( x 3 − x 2 y ) − ( 4 x y 2 − y 3 ) + ( 3 x 2 y − x y 2 ) ( x 3 − x 2 y ) − ( 4 x y 2 − y 3 ) + ( 3 x 2 y − x y 2 )

( x 3 − 2 x 2 y ) − ( x y 2 − 3 y 3 ) − ( x 2 y − 4 x y 2 ) ( x 3 − 2 x 2 y ) − ( x y 2 − 3 y 3 ) − ( x 2 y − 4 x y 2 )

In the following exercises, evaluate each polynomial for the given value.

Evaluate 8 y 2 − 3 y + 2 8 y 2 − 3 y + 2 when:

ⓐ y = 5 y = 5 ⓑ y = −2 y = −2 ⓒ y = 0 y = 0

Evaluate 5 y 2 − y − 7 5 y 2 − y − 7 when:

ⓐ y = −4 y = −4 ⓑ y = 1 y = 1 ⓒ y = 0 y = 0

Evaluate 4 − 36 x 4 − 36 x when:

ⓐ x = 3 x = 3 ⓑ x = 0 x = 0 ⓒ x = −1 x = −1

Evaluate 16 − 36 x 2 16 − 36 x 2 when:

ⓐ x = −1 x = −1 ⓑ x = 0 x = 0 ⓒ x = 2 x = 2

A painter drops a brush from a platform 75 feet high. The polynomial −16 t 2 + 75 −16 t 2 + 75 gives the height of the brush t t seconds after it was dropped. Find the height after t = 2 t = 2 seconds.

A girl drops a ball off a cliff into the ocean. The polynomial −16 t 2 + 250 −16 t 2 + 250 gives the height of a ball t t seconds after it is dropped from a 250-foot tall cliff. Find the height after t = 2 t = 2 seconds.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial −4 p 2 + 420 p . −4 p 2 + 420 p . Find the revenue received when p = 60 p = 60 dollars.

A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of p dollars each is given by the polynomial −4 p 2 + 420 p . −4 p 2 + 420 p . Find the revenue received when p = 90 p = 90 dollars.

Everyday Math

Fuel Efficiency The fuel efficiency (in miles per gallon) of a car going at a speed of x x miles per hour is given by the polynomial − 1 150 x 2 + 1 3 x − 1 150 x 2 + 1 3 x . Find the fuel efficiency when x = 30 mph x = 30 mph .

Stopping Distance The number of feet it takes for a car traveling at x x miles per hour to stop on dry, level concrete is given by the polynomial 0.06 x 2 + 1.1 x 0.06 x 2 + 1.1 x . Find the stopping distance when x = 40 mph x = 40 mph .

Rental Cost The cost to rent a rug cleaner for d d days is given by the polynomial 5.50 d + 25 5.50 d + 25 . Find the cost to rent the cleaner for 6 days.

Height of Projectile The height (in feet) of an object projected upward is given by the polynomial −16 t 2 + 60 t + 90 −16 t 2 + 60 t + 90 where t t represents time in seconds. Find the height after t = 2.5 t = 2.5 seconds.

Temperature Conversion The temperature in degrees Fahrenheit is given by the polynomial 9 5 c + 32 9 5 c + 32 where c c represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when c = 65 ° . c = 65 ° .

Writing Exercises

Using your own words, explain the difference between a monomial, a binomial, and a trinomial.

Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5.

Ariana thinks the sum 6 y 2 + 5 y 4 6 y 2 + 5 y 4 is 11 y 6 11 y 6 . What is wrong with her reasoning?

Jonathan thinks that 1 3 1 3 and 1 x 1 x are both monomials. What is wrong with his reasoning?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction

• Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis
• Publisher/website: OpenStax
• Book title: Elementary Algebra 2e
• Publication date: Apr 22, 2020
• Location: Houston, Texas
• Book URL: https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
• Section URL: https://openstax.org/books/elementary-algebra-2e/pages/6-1-add-and-subtract-polynomials

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• Adding a...

Adding and Subtracting Polynomials

In this tutorial, we’ll learn all about adding and subtracting polynomials. But first, there are a few fundamental concepts we need to look at.

Terms in a Polynomial

Here’s an example of a polynomial (with four terms).

The parts of the polynomial separated by the addition/subtraction symbols are called terms. And each term can be seen as having two parts – the coefficient and the variables (along with their exponents).

Here’s the first term in our polynomial above.

Note – You will frequently come across terms that appear to have no coefficients (or have variables without exponents). So remember, if the coefficient is not visible, it’s 1 \hspace{0.2em} 1 \hspace{0.2em} 1 . Same with the exponent. For example –

Terms are important because they hold the key when it comes to adding and subtracting polynomials.

Like/Unlike Terms

Like terms in a polynomial are those that have the same variable part (variables along with their exponents).

Here are a few examples of like terms. Focus on the variables parts (orchid and green).

• 2 x 2 , 5 x 2 , − 8 x 2 \hspace{0.2em} 2 {\color{Orchid} x^2} , \hspace{0.5em} 5 {\color{Orchid} x^2} , \hspace{0.5em} -8 {\color{Orchid} x^2} \hspace{0.2em} 2 x 2 , 5 x 2 , − 8 x 2
• 9 a 2 b , − a 2 b , 4 a 2 b \hspace{0.2em} 9 {\color{Teal} a^2 b} , \hspace{0.5em} - {\color{Teal} a^2 b} , \hspace{0.5em} 4 {\color{Teal} a^2 b} \hspace{0.2em} 9 a 2 b , − a 2 b , 4 a 2 b

Now, terms that are not like are known as “unlike terms”. Their variable parts (the variables themselves or their exponents) are different. For example (differences in red) –

• 4 x 2 , 3 x \hspace{0.2em} 4x^ {\color{Red} 2} , \hspace{0.5em} 3x \hspace{0.2em} 4 x 2 , 3 x
• 5 p 3 q 2 , 2 p 2 q 2 \hspace{0.2em} 5 p^ {\color{Red} 3} q^2, \hspace{0.5em} 2p^ {\color{Red} 2} q^2 \hspace{0.2em} 5 p 3 q 2 , 2 p 2 q 2
• 7 a b 2 c 3 , 3 a 2 c 3 \hspace{0.2em} 7a {\color{Red} b^2} c^3, \hspace{0.5em} 3a^ {\color{Red} 2} c^3 \hspace{0.2em} 7 a b 2 c 3 , 3 a 2 c 3

Adding and Subtracting Like Terms

The important thing about like terms is we can add (or subtract) them into a single term.

Let me explain. Here we have two like terms being added together.

Now, there are two steps to adding like terms.

Step 1.  Take the coefficients and add/subtract them.

Step 2.  Copy the variable part.

Nothing too difficult, right? We can also subtract like terms using the same steps.

Alright, let’s do a couple of examples.

Solution ( i \hspace{0.2em} i \hspace{0.2em} i )

Just like we did earlier, we’ll bring the coefficients together, simplify them, and copy down the variables.

Solution ( i i \hspace{0.2em} ii \hspace{0.2em} ii )

Again, the same two steps.

Great! We are all set to start adding and subtracting polynomials.

We'll go with addition first. Subtraction would require only a slight adjustment.

Adding Polynomials

• Write all the polynomials one after the other connected with the plus sign.
• Group the like terms together.
• Simplify each group of like terms into one term.

Let’s apply these steps to solve a few polynomial addition problems.

Add and simplify.

Step 1.  We write the two polynomials one after the other with a plus sign between them.

Like terms are in the same color.

Step 2.  Next, we bring the like terms together and simplify each group (using what we learned in the last section).

We can’t simplify any further (can’t combine unlike terms). So that’s the answer.

Nothing really different in this example. Using the same two steps from above, we get –

That’s it.

Vertical (Column-wise) Addition

Instead of writing all the polynomials in one row, you can also arrange their terms in columns and add.

• Write all the polynomials one below the other such that each group of like terms falls in a separate column.
• Add the terms (with their signs) in each column.

The following examples should help you understand this better.

Step 1.  We write the polynomials one below the other such that each group of like terms falls in a separate column.

Also, draw a horizontal line below the polynomials as shown below.

Step 2.  Add the terms (with their signs) in each column and write the sum below the line.

So the sum of the two polynomials is 13 a 2 b + 5 a b 2 + 2 a b + 7 \hspace{0.2em} 13a^2b + 5ab^2 + 2ab + 7 \hspace{0.2em} 13 a 2 b + 5 a b 2 + 2 ab + 7 .

Again, the same two steps. Just make sure only like terms go into the same column.

In our final answer, we don’t need to write the second term (it’s 0 \hspace{0.2em} 0 \hspace{0.2em} 0 ). The sum is − 10 x 3 y 2 + 3 x y 2 + 2 x y \hspace{0.2em} -10x^3y^2 + 3xy^2 + 2xy \hspace{0.2em} − 10 x 3 y 2 + 3 x y 2 + 2 x y

Subtracting Polynomials

Subtracting a polynomial is the same as inverting the sign of each of its terms ( − \hspace{0.2em} - \hspace{0.2em} − ↔ + \hspace{0.2em} + \hspace{0.2em} + ), and adding the polynomial.

Let me explain through this example.

Subtract the second polynomial from the first.

Step 1.  We write the two polynomials one after the other with a minus sign between them.

Now it’s important to write the second polynomial inside parentheses because it’s the whole polynomial we are subtracting and not just the first term.

Step 2.  Open the parentheses and distribute the minus sign across the second polynomial. In other words, we flip the sign of each term in the second polynomial from + \hspace{0.2em} + \hspace{0.2em} + to − \hspace{0.2em} - \hspace{0.2em} − and the other way around. (Shown above)

Step 3.  Group the like terms together and simplify.

Again, we start by writing the polynomials together with a minus sign between them.

I repeat, make sure the second polynomial is enclosed within parentheses. And that you flip the signs when opening the parentheses.

And now, it’s time to simplify each group of like terms.

Cool! Now let’s do a couple of examples using the vertical method – column-wise subtraction.

Vertical (Column-wise) Subtraction

• Write the two polynomials one below the other such that each group of like terms falls in a separate column.
• Flip the signs of each term ( − \hspace{0.2em} - \hspace{0.2em} − ↔ + \hspace{0.2em} + \hspace{0.2em} + ) in the second row (of the polynomial being subtracted).

Step 1.  Write the second polynomial below the first such that each pair of like terms falls in a separate column.

Again, draw a horizontal line below the polynomials as shown below.

Step 2.  Flip the signs of each term in the second row.

Remember, if a term doesn’t have a visible sign, its sign is + \hspace{0.2em} + \hspace{0.2em} + (and so, we change it to − \hspace{0.2em} - \hspace{0.2em} − ).

Step 3.  Add the terms (with their news signs) in each column and write the sum below the line.

So, the difference is − 7 x 2 − 9 x − 3 \hspace{0.2em} -7x^2 - 9x - 3 \hspace{0.2em} − 7 x 2 − 9 x − 3

The same three steps and we’ll get the answer.

So p 3 q 2 + 3 p 2 q − 5 p q − 1 \hspace{0.2em} p^3q^2 + 3p^2q - 5pq - 1 \hspace{0.2em} p 3 q 2 + 3 p 2 q − 5 pq − 1 is our answer.

And that brings us to the end of this tutorial on adding and subtracting polynomials. Until next time.

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Adding And Subtracting Polynomials

In these lessons, we will look at how to add and subtract polynomials.

Related Pages Algebra Terms Algebraic Expressions Adding & Subtracting Functions More Algebra Lessons Grade 7 Math Lessons

The following diagram shows examples of adding and subtracting polynomials. Scroll down the page for more examples and solutions on how to add and subtract polynomials.

Adding Polynomials

Adding polynomials involves combining like terms .

Example: Add the polynomials 5x – 2 + y and –3y + 5x + 2

Solution: 5x – 2 + y + (–3y + 5x + 2) = 5x + 5x + y – 3y – 2 + 2 = 10x – 2y

Example: Find the sum of –7x 3 y + 4x 2 y 2 – 2 and 4x 3 y + 1 – 8x 2 y 2

Solution: –7x 3 y + 4x 2 y 2 – 2 + 4x 3 y + 1 – 8x 2 y 2 = –7x 3 y + 4x 3 y + 4x 2 y 2 – 8x 2 y 2 – 2 + 1 = –3x 3 y – 4x 2 y 2 – 1

Subtracting Polynomials

To subtract polynomials, remember to distribute the – sign into all the terms in the parenthesis.

Example: Simplify –4x + 7 – (5x – 3)

Solution: –4x + 7 – (5x – 3) = –4x + 7 – 5x + 3 = –9x + 10

Example: Simplify (5x 2 + 2) – (– 4x 2 + 7) + (– 3x 2 – 5)

Solution: (5x 2 + 2) – (– 4x 2 + 7) + (– 3x 2 – 5) = 5x 2 + 2 + 4x 2 – 7 – 3x 2 – 5 = 5x 2 + 4x 2 – 3x 2 + 2 – 7 – 5 = 6x 2 – 10

How To Add And Subtract Polynomials?

To add polynomials

• Combine like terms.
• Write in descending order.
• (4x 2 + 8x - 7) + (2x 2 - 5x - 12)
• (5 + 24y 3 - 7y 2 ) + (-6y 3 + 7y 2 + 5)
• (t 2 - t + 5) + (7t 2 - 4t - 20)

To subtract polynomials

• Rewrite the subtraction as addition.

To change subtraction to addition, we must add the opposite, or additive inverse. 2. Combine like terms. 3. Write in descending order.

• (4x 2 + 8x - 7) - (2x 2 - 5x - 12)
• (6x 4 + 3x 3 - 1) - (4x 3 - 5x + 9)
• (1.5y 3 + 4.8y 2 + 12) - (y 3 - 1.7y 2 + 2y)

Examples Of Adding And Subtracting Polynomials

• (2x 5 - 6x 3 - 12x 2 - 4) + (-11x 5 + 8x + 2x 2 + 6)
• (-9y 3 - 6y 2 - 11x + 2) - (-9y 4 - 8y 3 + 4x 2 + 2x)

Adding polynomials and subtracting polynomials is essentially combining like terms of polynomial expressions. When adding and subtracting polynomials, they can either be arranged vertically or grouped according to degree. A knowledge of polynomial vocabulary is important before adding and subtracting polynomials. Multiplying monomials and binomials is another type of operation with polynomials.

• (4x 2 - 3x + 2) + (5x 2 + 2x - 7)
• (5x 3 + 7x 2 - x) + (8x 3 + 4x - 5)
• (8x 2 + 2x) - (10x 2 + 2x - 9)
• -(6x 3 - 4x) - (2x 3 + x 2 -2x)

• (x 2 + 4x + 5) + (6x + 3)
• 2(x 4 + 5x) - 6(x 4 + 8x - 3)

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Addition and Subtraction of Polynomials

Addition of Polynomials

The sum of two specific numbers can be written as a third specific number. Given the specific numbers 2 and 3, we can write their sum as 2 + 3 , or as 5 .

The sum of the literal numbers a and b can merely be indicated as a + b. The a and b are called terms of the sum.

Terms that have identical literal factors are called like terms :  2abc , 3bac , - 10cba are like terms. On the other hand, 2abc and 3abd are unlike terms , since 2abc has c as a factor while 3abd does not. The numerical coefficients of the terms do not affect whether the terms are like or unlike.

When the terms to be added are like terms, such as 5a and 7a , the sum 5a + 7a can be simplified by use of the distributive law of multiplication.

Using the distributive law of multiplication, we have

     5a + 7a = (5 + 7)a = 12a

Also 4a + a - 8a = (4+1)a - 8a

or 4a + a - 8a = (4 + 1 - 8)a

It is important to realize that

     2a + 4a = (2 + 4)a = 6a

and 5a - 3a = (5-3)a = 2a (not just 2)

When we add polynomials, we combine only like terms in the polynomials

Let's see how our Polynomial solver simplifies this and similar problems. Click on "Solve Similar" button to see more examples solved step by step.

Example Add 3a - 5b and  - 2a+2b

Solution (3a - 5b) + (-2a + 3b) = 3a - 5b - 2a + 3b

  = (3a - 2a) + (-5b + 3b)

  = (3-2)a + (-5 + 3)b = a - 2b

Example Add 3a - 2b + c and 6a + 4b - 5c

Solution (3a + 2b + c) + (6a + 4b -5c) = 3a - 2b +c +6a + 4b -5c

 = (3a + 6a) + (-2b + 4b) + (c -5c)

= (3 + 6)a + (-2 + 4)b + (1-5)c

= 9a + 2b -4c

An easy way to find the sum of polynomials is to write the polynomials in rows one under the other, so that like terms are in the same column. This is similar to addition of

Specific numbers, when we write specific numbers in rows so that the units. tens, hundreds, and so forth are in separate columns.

Example Find the sum of the following polynomials

       2ab -6c +d 3c - 5d and 2d - 4ab +4c

       (2ab - 6c +d) + (3c - 5d) + (2d - 4ab + 4c) = -2ab + c - 2d

Let's see how our Polynomial solver shows all the solution steps for this and similar problems. Click on "Solve Similar" button to see more examples.

Subtraction of Polynomials

In algebraic language we symbolize the subtraction of b from a by a - b , which is the same as a + (-b) . That is. to subtract b from a, we add the additive inverse (the negative) of b to a .

The additive inverse of + 6x is - 6x . That is, - (+6x) = - 6x .

The additive inverse of - 10y is + 10y . That is, - (-10y) = + 10y

When the terms to be subtracted are like terms. the difference can be simplified using the distributive law of multiplication.

Example 1. Subtract (3a) from (8a)

          (8a) - (3a) = 8a - 3a = (8 - 3)a = 5a

2. Subtract (-3a) from (8a) .

          (8a) - (-3a) = 8a + 3a = (8 + 3)a = 11a

3. Subtract (3a) from (-8a) .

          (-8a) - (3a) = -8a - 3a = (-8 -3)a = -11a

4. Subtract (-3a) from (-8a) .

          (-8a) - (-3a) = -8a + 3a = (-8 +3)a = -5a  

To subtract one polynomial, called the subtrahend , from another polynomial, called the minuend , add the minuend to the additive inverse of the subtrahend and combine like terms.

The additive inverse of a polynomial is the additive inverse of every term in the polynomial.

The additive inverse of 5a - 6b + 8 is -5a + 6b -8 That is,

       -(5a - 6b + 8) = -5a + 6b - 8

Example Subtract (3a - 5b) from (6a - 7b)

Solution (6a - 7b) - (3a - 5b) = 6a - 7b - 3a + 5b

= (6a - 3a) + (-7b + 5b)

= (6 - 3)a + (-7 +5)b

Let's see how our polynomial solver simplifies this problem step by step. Click on "Solve Similar" button to see more examples.

Example Subtract (3a - 2b + 5) from (8a + 6b -2)

Solution (8a + 6b -2) - (3a -2b +5) = 8a + 6b -2 -3a +2b -5

  = (8a -3a) + (6b +2b) + (-2 -5)

  = (8 -3)a + (6 +2)b + (-2 -5)

  = 5a +8b -7

We can subtract polynomials more simply by writing them in rows. Write the minuend in the first row and the subtrahend in the second row, so that like terms are in the same column. The additive inverse of a polynomial is the same polynomial with the signs of the terms changed. Thus we change the signs of each term in the subtrahend. write the new signs in circles above the original signs, and add like terms using the new signs.

Example From 7ab -2c +8 subtract 8ab -5c +4

Solution   

       (7ab -2c +8) - (8ab -5c +4) = -ab +3c +4

Let's see how our Polynomial simplifier shows step by step solutions to  this and similar problems. Click on "Solve Similar" button to see more examples.

Example Subtract 2x - 3y - 6 from 4x - 3y +10

Solution  

Since 0 , y = 0 , that term does have to be written.

       (4x - 3y + 10) - (2x - 3y - 6) = 2x +16

Example Subtract 6ab + 2c -4 from 8ab -2b +3

       (8ab -2b +3) - (6ab + 2c - 4) = 2ab - 2b + 7 - 2c

Math Topics

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Worksheet on Adding and Subtracting Polynomials

Practice the problems given in the worksheet on adding and subtracting polynomials. The questions are based on different types of word problems on addition and subtraction of polynomials.

1. Solve the following subtraction:

(i) Take – (3/2)a + b – c from (1/2)a – (1/3)b – (3/2)c

(iii) Take 3a + 2b – 6c from 8a – 4b – 2c

2. (i) What should be added to 3m to get 5m?

(ii) What should be added to -p to get 4p?

(iii) What should be added to x - y to get 2x + y?

3.  From the sum of a + b – 2c and 2a – b + c, subtract a + b + c.

4.  Subtract the sum of p + q and p – r from the sum of p – 2r and p + q + r.

6. From the sum of a - b + 11 and –b – 9, subtract 2a - 3b - 1.

Answers for the worksheet on adding and subtracting polynomials are given below to check the exact answers of the above word problems.

1. (i) 2a – (4/3)b – (1/2)c

(iii) 5a – 6b + 4c

(iii) x + 2y

3. 2a – b – 2c

● Terms of an Algebraic Expression - Worksheet

Worksheet on Types of Algebraic Expressions

Worksheet on Degree of a Polynomial

Worksheet on Addition of Polynomials

Worksheet on Subtraction of Polynomials

Worksheet on Addition and Subtraction of Polynomials

Worksheet on Multiplying Monomials

Worksheet on Multiplying Monomial and Binomial

Worksheet on Multiplying Monomial and Polynomial

Worksheet on Multiplying Binomials

Worksheet on Dividing Monomials

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Add & Subtract Polynomials Worksheets

How Do You Add and Subtract Polynomials? Polynomials are one of the most significant parts of algebra. Students are aware of linear, quadratic, and cubic equations as they are the most commonly used ones. Any algebraic equations with the exponent on the variable higher than three are termed as polynomials. The polynomials consist of variables, co-efficient, and addition or subtraction operations. Adding and subtracting polynomials is not complicated. All you have to do is put the like terms together and start simplifying it until it cannot be further reduced or simplified. Example: Add 3x 4 + 2x + 4x 2 -10 + 5x 3 and 10x 3 + 5x - 2x 2 +4 To add these two polynomials, you have to write it down in the form of a sum; x 4 + 2x + 4x 2 -10 + 5x 3 + (10x 3 + 5x - 2x 2 +4) Start arranging the terms in a way that the exponent on the variables in descending order. x 4 +5x 3 + 10x 3 + 4x 2 - 2x 2 + 2x + 5x - 10 + 4 Now you see, by placing the values in a way that the exponents are in descending order, you have gathered the like terms together. x 4 +15x 3 +2x 2 + 7x - 6

Basic Lesson

Demonstrates how to add & subtract common polynomials. Change the signs of all the terms being subtracted. Change the subtraction signs to addition signs.

Intermediate Lesson

Explores how to solve Polynomials operations with unlike terms.

Independent Practice 1

Contains 20 add & subtract polynomials problems. The answers can be found below.

Independent Practice 2

Features another 20 Add & Subtract Polynomials problems.

Homework Worksheet

Add & Subtract Polynomials problems for students to work on at home. Example problems are provided and explained.

10 Add & Subtract Polynomials problems. A math scoring matrix is included.

Homework and Quiz Answer Key

Answers for the homework and quiz.

Lesson and Practice Answer Key

Answers for both lessons and both practice sheets.

Question: What Greek math whiz noticed that the morning star and evening star were one and the same, in 530 B.C.? Answer: Pythagoras.

Adding and Subtracting Polynomials Worksheet 25 question pdf with answer key

Students will practice adding and subtracting polynomials.

Example Questions

Other Details

This is a 4 part worksheet:

• Part I Model Problems
• Part II Practice
• Part III Challenge Problems
• Part IV Answer Key
• How to Add and Subtract Polynomials
• Polynomial Equations

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Polynomials: Adding & Subtracting

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Number Line

• (2x^2-1)+(5x-6)
• (x^2+2x-1)-(2x^2-3x+6)
• (x^2+2x-1)\cdot(2x^2-3x+6)
• long\:division\:\frac{x^{3}+x^{2}}{x^{2}+x-2}
• factor\:5a^2-30a+45
• What is a polynomial?
• A polynomial is a mathematical expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, and multiplication. Polynomials are often written in the form: a₀ + a₁x + a₂x² + a₃x³ + ... + aₙxⁿ, where the a's are coefficients and x is the variable.
• How do you identify a polynomial?
• To identify a polynomial check that: Polynomials include variables raised to positive integer powers, such as x, x², x³, and so on. Polynomials involve only the operations of addition, subtraction, and multiplication. Polynomials include constants, which are numerical coefficients that are multiplied by variables.
• What are the types of polynomials terms?
• The types of polynomial terms are: Constant terms: terms with no variables and a numerical coefficient. Linear terms: terms that have a single variable and a power of 1. Quadratic terms: terms that have a single variable and a power of 2. Cubic terms: terms that have a single variable and a power of 3. Higher-order terms: terms that have a single variable and a power of 4 or higher. Mixed terms: terms that have multiple variables with different powers.
• How do you calculate a polynomial?
• To calculate a polynomial, substitute a value for each variable in the polynomial expression and then perform the arithmetic operations to obtain the result.
• What are monomial, binomial, and trinomial?
• A monomial is a polynomial with a single term, a binomial is a polynomial with two terms, and a trinomial is a polynomial with three terms.

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