multiplying polynomials common core algebra 1 homework

Big Ideas Math - Algebra 1, A Common Core Curriculum

By larson, ron; boswell, laurie, chapter 7 - polynomial equations and factoring - 7.2 - multiplying polynomials - exercises - page 369: 6, work step by step, update this answer.

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Multiplying Polynomials Worksheet and Answer Key

Students will practice multiplying polynomials including multiplying a polynomial by monomial , foiling binomials and multiplying polynomials by polynomials .

Example Questions

Directions: multiply the polynomials below

Other Details

This is a 4 part worksheet:

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

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Common Core Algebra I Math (Worksheets, Homework, Lesson Plans)

Related Topics: Common Core Math Resources, Lesson Plans & Worksheets for all grades Common Core Math Video Lessons, Math Worksheets and Games for Algebra Common Core Math Video Lessons, Math Worksheets and Games for all grades

Looking for video lessons that will help you in your Common Core Algebra I math classwork or homework? Looking for Common Core Math Worksheets and Lesson Plans that will help you prepare lessons for Algebra I students?

The following lesson plans and worksheets are from the New York State Education Department Common Core-aligned educational resources. Eureka/EngageNY Math Algebra I Worksheets.

These Lesson Plans and Worksheets are divided into five modules.

Algebra I Homework, Lesson Plans and Worksheets

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Polynomial Multiplication Worksheets

Polynomials are algebraic expressions that consist of several terms. Each term often contains different powers of the same variable(s). The name often haunts students, but the problems associated with the, especially operations, are no more difficult than working with larger numbers in operations. It all begins and ends with the ability of the student to stay organized through the process and being able to spot like terms. This particular operation is used often in the construction industry to determine the desired dimensions of a structure that will be fabricated. It is also essential when constructing and engineering any type of curved structure. These worksheets and lessons help students learn how to multiply polynomials.

Aligned Standard: HSA-APR.A.1

• Polynomial Product Step-by-step Lesson -We work through your first polynomial product and help you solve them as a series.
• Guided Lesson - These are three problems on polynomials that are very common to see. You might even recognize the common pattern that is used for these.
• Guided Lesson Explanation - Work with the concept of (FOIL) First-Outside-Inside-Last for solving these.
• Practice Worksheet - I threw a couple in there with three terms to give you a bit of a challenge.
• Matching Worksheet - Match the polynomial products to their final outcome.
• Multiplying Polynomials Five Pack - A rapid fire splash of problems for you to practice with. These problems are the standard level you will see.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

Once again, get the kids in the habit of treating polynomials like a three or more part equation.

• Homework 1 - In algebra when we use the distributive property, we are expanding or distributing.
• Homework 2 - Now, multiply the third bracket with the product of first and the second bracket.
• Homework 3 - We will multiply both the parentheses.

Practice Worksheets

Remind students that spacing these problems is critical when solving them.

• Practice 1 - Find the product of the polynomials.
• Practice 2 - (5x + 2) (5x - 5)
• Practice 3 - What is the final value left behind?

Math Skill Quizzes

I threw a couple of curve balls in here to keep kids on their toes.

• Quiz 1 - Find the products.
• Quiz 2 - We add exponents in this quiz.
• Quiz 3 - The zero value always throws them off.

How to Multiply Polynomials

2 x 2 + 3y +4z

We all have seen an expression or a mathematical statement like the one above. Such combinations of variables (letters like x, y, z), constants (numbers like 2, 3, 4) and exponents (like 2 in x 2 ) are known as polynomials. Polynomials contain operators like: multiplication, addition, subtraction, and positive exponents. But they don't have division operators or negative exponents.

Here, we are going to focus on the multiplication operator with polynomials. There are a couple of things you need to keep in mind when multiplying polynomials with each other:

Step 1 - Multiply each term in one polynomial by each term in the other polynomial.

Step 2 - Combine those terms and simplify if needed.

Let's begin with the easiest of the bunch and work our way up the spectrum.

Monomial With Monomial (1 Term Times 1 Term) - To multiply a monomial with monomial, we first multiply the coefficients (multipliers) then the variables and find the result.

(3 . 4) (a . a) (b) : Place like terms together.

12 a 2 b: Note that when multiplying variables, we add their exponents.

Monomial With Binomial (1 Term Times 2 Terms) - Multiply the single term with each of the two terms of binomial. For example,

(a 2 - a) 2b

(a 2 -. 2b) - (a. 2b)

2a 2 b - 2ab

Binomial With Binomial (2 Terms Times 2 Terms) - This one is a bit longer than the first two types of polynomial multiplication. In this multiplication, each of the two terms in one binomial is multiplied with each of the two terms in the other binomial.

To elaborate with an example, (a + b) (c +d)

Taking the first binomial: a.c + a.d + b.c + b.d = ac + ad + bc +bd

You can multiply the binomials in any order, make sure that each of the first two terms is being multiplied by each second term of the binomials.

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Multiplying Polynomials for Algebra 1

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Description

Common Core State Standards Math

Cluster: Perform arithmetic operations on polynomials

Standard: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Degree of Alignment: 2 Strong (1 user)

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Unit 7 – Polynomials

Introduction to Polynomials

LESSON/HOMEWORK

LECCIÓN/TAREA

LESSON VIDEO

EDITABLE LESSON

EDITABLE KEY

SMART NOTEBOOK

Adding and Subtracting Polynomials

Multiplying Polynomials

More Work Multiplying Polynomials

Factoring Polynomials

Conjugate Binomials

Factoring Trinomials

Complete Factoring

Recognizing Structure to Factor

Factoring Challenging Trinomials

Unit Review

Unit 7 Review

UNIT REVIEW

REPASO DE LA UNIDAD

EDITABLE REVIEW

Unit 7 Assessment – Form A

EDITABLE ASSESSMENT

Unit 7 Assessment – Form B

Unit 7 Exit Tickets

Unit 7 Mid-Unit Quiz – Form A

U07.AO.01 – Optimizing the Volume of an Open Box

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Common Core State Standards Initiative

High School: Algebra

Standards in this domain:, ccss.math.content.hsa.introduction introduction, expressions..

An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function. Conventions about the use of parentheses and the order of operations assure that each expression is unambiguous. Creating an expression that describes a computation involving a general quantity requires the ability to express the computation in general terms, abstracting from specific instances.

Reading an expression with comprehension involves analysis of its underlying structure. This may suggest a different but equivalent way of writing the expression that exhibits some different aspect of its meaning. For example, p + 0.05 p can be interpreted as the addition of a 5% tax to a price p . Rewriting p + 0.05 p as 1.05 p shows that adding a tax is the same as multiplying the price by a constant factor.

Algebraic manipulations are governed by the properties of operations and exponents, and the conventions of algebraic notation. At times, an expression is the result of applying operations to simpler expressions. For example, p + 0.05 p is the sum of the simpler expressions p and 0.05 p . Viewing an expression as the result of operation on simpler expressions can sometimes clarify its underlying structure.

A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic expressions, perform complicated algebraic manipulations, and understand how algebraic manipulations behave.

Equations and inequalities.

An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. An identity, in contrast, is true for all values of the variables; identities are often developed by rewriting an expression in an equivalent form.

The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs of numbers, which can be plotted in the coordinate plane. Two or more equations and/or inequalities form a system. A solution for such a system must satisfy every equation and inequality in the system.

An equation can often be solved by successively deducing from it one or more simpler equations. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions.

Some equations have no solutions in a given number system, but have a solution in a larger system. For example, the solution of x + 1 = 0 is an integer, not a whole number; the solution of 2 x + 1 = 0 is a rational number, not an integer; the solutions of x 2 - 2 = 0 are real numbers, not rational numbers; and the solutions of x 2 + 2 = 0 are complex numbers, not real numbers.

The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = (( b 1 + b 2 )/2) h , can be solved for h using the same deductive process. Inequalities can be solved by reasoning about the properties of inequality. Many, but not all, of the properties of equality continue to hold for inequalities and can be useful in solving them.

Connections to Functions and Modeling. Expressions can define functions, and equivalent expressions define the same function. Asking when two functions have the same value for the same input leads to an equation; graphing the two functions allows for finding approximate solutions of the equation. Converting a verbal description to an equation, inequality, or system of these is an essential skill in modeling.

Algebra Overview

• Seeing Structure in Expressions
• Interpret the structure of expressions
• Write expressions in equivalent forms to solve problems

Arithmetic with Polynomials and Rational Functions

• Perform arithmetic operations on polynomials
• Understand the relationship between zeros and factors of polynomials
• Use polynomial identities to solve problems
• Rewrite rational functions

Creating Equations

• Create equations that describe numbers or relationships

Reasoning with Equations and Inequalities

• Understand solving equations as a process of reasoning and explain the reasoning
• Solve equations and inequalities in one variable
• Solve systems of equations
• Represent and solve equations and inequalities graphically

Mathematical Practices

• Make sense of problems and persevere in solving them.
• Reason abstractly and quantitatively.
• Construct viable arguments and critique the reasoning of others.
• Model with mathematics.
• Use appropriate tools strategically.
• Attend to precision.
• Look for and make use of structure.
• Look for and express regularity in repeated reasoning.

Arithmetic with Polynomials & Rational Expressions

Creating equations ✭, reasoning with equations & inequalities.

• Standards for Mathematical Practice
• How to read the grade level standards
• Introduction
• Counting & Cardinality
• Operations & Algebraic Thinking
• Number & Operations in Base Ten
• Measurement & Data
• Number & Operations—Fractions¹
• Number & Operations in Base Ten¹
• Number & Operations—Fractions
• Ratios & Proportional Relationships
• The Number System
• Expressions & Equations
• Statistics & Probability
• The Real Number System
• Quantities*
• The Complex Number System
• Vector & Matrix Quantities
• Arithmetic with Polynomials & Rational Expressions
• Creating Equations*
• Reasoning with Equations & Inequalities
• Interpreting Functions
• Building Functions
• Linear, Quadratic, & Exponential Models*
• Trigonometric Functions
• High School: Modeling
• Similarity, Right Triangles, & Trigonometry
• Expressing Geometric Properties with Equations
• Geometric Measurement & Dimension
• Modeling with Geometry
• Interpreting Categorical & Quantitative Data
• Making Inferences & Justifying Conclusions
• Conditional Probability & the Rules of Probability
• Using Probability to Make Decisions
• Courses & Transitions
• Mathematics Glossary
• Mathematics Appendix A

Multiplying Polynomials

Multiplication is one of the arithmetic operations which can be applied to polynomials. Multiplying polynomials is one of the simplest things in algebra. Polynomials can be easily multiplied by using their rules. When multiplying polynomials we multiply coefficients together and variables together. In this chapter, we will discuss the multiplication of polynomials, their rules, and the steps to multiply polynomials.

Multiplication of Polynomials

Polynomial multiplication is a method for multiplying two or more polynomials together. The terms of the 1 st polynomial are multiplied with the 2 nd polynomial to get the resultant polynomial. Based on the types of polynomials we use, there are different ways of multiplying them. The rules for the multiplication of polynomials are different for each type of polynomial . To multiply polynomials, the coefficient is multiplied with a coefficient, and the variable is multiplied with a variable.

Multiplying Polynomials with Exponents

When the polynomials are multiplied it is possible they can be monomial, binomial, or trinomial. In order to multiply any two polynomials the steps used are:

• Multiply the coefficients
• Multiply the variables using exponent rules as per the requirement.

Let us understand how to multiply polynomials with exponents using an example.

Example: Multiply 2x 3 with 3x 2

We will follow the same procedure for multiplying polynomials with exponents as we had done above.

• Step 1: First we will multiply the coefficients i.e., 2 × 3 = 6
• Step 2: Next, we will multiply the variables but in this case, the powers of both the variables will be added as per the rules of exponents i.e., x 3 × x 2 = x 5

The final answer is 2x 3 × 3x 2 = 6x 5

Multiplying Polynomials with Different Variables

It is possible to multiply polynomials with different variables too. The steps to multiply polynomials with different variables are:

• Multiply the variables and use rules of exponents wherever necessary.

Example: Multiply 5x 2 with 3y.

• Step 1: We will first multiply the coefficients of both the polynomials i.e., 5 × 3= 15
• Step 2: Since the above polynomials have two different variables, they cannot be multiplied. Hence, we will keep them the same.

The final answer is 5x 2 × 3y = 15x 2 y

How to Multiply Monomials?

Monomials are polynomials having just one term, consisting of a variable and its coefficient. Hence the steps to determine the product of two or three monomials follow the same steps as we learned above. It is possible to multiply monomials more than three too using the same steps we will learn for the below examples.

Multiplication of Two Monomials

When multiplying monomials, we need to follow certain rules similar to multiplying polynomials. Let us understand by taking two monomials, 3x and 2x.

• Step 1 : In the above monomials, the common variable is x. We will multiply the variable with the variable. Hence, we get x × x = x 2 .
• Step 2 : In the next step, we will multiply the coefficients of both the monomials to get 2 × 3 = 6. Thus, multiplying the polynomials 2x and 3x gives 6x 2 as the result.

Multiplication of Three Monomials

To multiply three monomials, we will use the same method as that used for multiplying two monomials. Let us understand the method with an example.

Example: Multiply 2x, 3y, and 6z.

• Step 1 : First we will multiply the variables together i.e., x × y × z = xyz
• Step 2 : Next we will multiply the coefficients of all the three terms i.e., 2 × 3 × 6 = 36

Thus, the multiplication result can be shown as 2x × 3y × 6z = 2 × 3 × 6 × x × y × z = 36xyz

How to Multiply Binomials?

Binomials are a particular kind of polynomials consisting of only two terms. They can be multiplied in two ways:

• Distributive Property

Multiplying Binomials by Distributive Property

For multiplying binomials, we use the distributive property. Let's multiply a binomial (a+b) with another binomial (c+d).

• Step 1: Write both the binomials together i.e., (a + b)(c + d)
• Step 2: Out of the two brackets, keep one bracket constant, let's say (c + d).
• Step 3: Now multiply each and every term from the other bracket i.e., (a + b) with (c + d).

Multiply (2x+3)(4x+5)

The above polynomials can be solved as: (2x + 3)(4x + 5) = 2x(4x + 5) + 3(4x + 5) ⇒ 8x 2 + 10x + 12x + 15 ⇒ 8x 2 + 22x + 15

Multiplying Binomials by Box Method

Two binomials can also be multiplied using the box method. The terms are written across a box and their corresponding products are written inside the box.

Multiply (x + 7) with (x + 3) Solution: Let's write the polynomials (x + 7) horizontally and (x + 3) vertically. Take the sign with its corresponding term on the right. After multiplying the corresponding terms, we get:

Thus, the above multiplication method is known as the box multiplication of two binomials. We now have (x 2 + 7x + 3x + 21) as the sum. Thus, the final product will be (x 2 + 10x + 21).

How to Multiply a Monomial with a Binomial?

As we did above, to multiply a monomial with a binomial, we have to use the distributive property. Let's say monomial a has to be multiplied with binomial (b + c). By distributive property, the above product can be written as: a(b + c) = ab + ac.

Multiply 3y with (5x + 2z)

3y(5x + 2z) = 3y × 5x + 3y × 2z

⇒ (3 × 5 × y × x) + (3 × 2 × y × z) = 15yx + 6yz

Topics Related to Multiplying Polynomials

Check out these interesting articles to learn more about multiplying polynomial and its related topics.

• Multiplying Polynomials Calculator
• Multiplying Binomials Calculator
• Polynomial Calculator

Tips to Remember

• When multiplying polynomials, the coefficient will be multiplied with a coefficient and the variable will be multiplied with a variable.
• Polynomials can also be solved using the distributive property, box method, or grid method.
• When multiplying polynomials with exponents, the rules of exponents have to be used.

Multiplying Polynomials Examples

Example 1: Simplify xz(x 2 + z 2 ) by using rules of multiplication of polynomials.

Solution: xz(x 2 + z 2 ) = (xz × x 2 ) + (xz × z 2 )

⇒ x 3 z + xz 3

Therefore, the product is x 3 z + xz 3

Example 2: Find the product: (2x + 3y)(4x - 5y)

Solution: By using distributive property for multiplying polynomials, we get

2x(4x -5y) + 3y(4x - 5y) = 8x 2 - 10xy + 12xy -15y 2

⇒ 8x 2 + 2xy - 15y 2

Therefore, the product is 8x 2 + 2xy - 15y 2

Example 3: A cuboid has sides measuring 2y, 3x and 5z as its length, breadth, and height respectively. Find the volume of the cuboid.

Solution: As we know, the volume of cuboid = length × breadth × height. Here, all the side lengths are given in the form of monomials, so by applying rules for multiplying monomials, we get,

⇒ Volume = 2y × 3x × 5z = 30xyz

Therefore, the volume of the cuboid is 30xyz cubic units.

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Practice Questions on Multiplying Polynomials

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FAQs on Multiplying Polynomials

How do you multiply three polynomials.

Multiplication of three polynomials is a two-step process that involves the following two steps:

• Multiplication of coefficients
• Multiplication of the variables using Laws of Exponents as and when required.

Let's take an example to understand the multiplication of three polynomials. Example: Multiply (3m+2), 4n 2 , and 7p.

• The above given three polynomials are written as (3m+2)× 4n 2 × 7p
• By using distributive property of polynomial multiplication we get, ((3m× 4n 2 )+(2× 4n 2 ))× 7p = (12mn 2 + 8n 2 )7p = 84mn 2 p + 56n 2 p

Thus, the above multiplication can be shown as (3m+2)× 4n 2 × 7p = 84mn 2 p + 56n 2 p.

How can we Multiply Polynomials Using the Box Method?

Two or more polynomials can be multiplied using the box method. The terms are written across a box and their corresponding products are written within the box. Example: (3x 2 + 2x + 4)(4x + 5)

3x 2 +2x+4 will be written on the vertical side of the box while 4x+5 will be written on the horizontal side of the box, or vice-versa. Then, first, we will multiply 3x 2 by 4x, then 3x 2 by 5, and write the products in the corresponding section of the box. Secondly, we will multiply 2x by 4x and 2x by 5 and write down the products. The final column of the box is filled by multiplying 4 by 4x and 4 by 5. At last, we will add all six terms obtained to get the final answer. Therefore, the result of the multiplication of both the polynomials is (12x 3 +23x 2 +26x+20).

How do you Multiply Binomials Using the Grid Method?

The steps to multiply polynomials by a box method or the grid method is as follows: Example: (x + 6)(2x + 3)

x+6 will be written on the vertical side of the box while 2x+3 will be written on the horizontal side of the box, or vice-versa. Multiply each term with the respective terms. Therefore, the product which we get is (2x 2 + 15x + 18).

How Many Methods are there for Multiplying Polynomials?

There are two methods for multiplying polynomials:

What does FOIL Stand for in Multiplying Binomials?

FOIL stands for First, Outer, Inner Last in multiplying binomials. The binomials are multiplied as:

• Step 1: Multiply the first term of each binomial.
• Step 2: Now multiply the outer term of each binomial.
• Step 3: Once this is done, now multiply the inner terms of the binomials.
• Step 4: Now the last terms are multiplied.
• Step 5: Once all the above four steps are done, the products obtained as each step are added, like terms are combined and the answer is simplified.

What is the Best Method for Multiplying Polynomials?

The best method for multiplying polynomials is the distributive property of multiplying polynomials. The steps to multiply a polynomial using the distributive property are:

• Step 1: Write both the polynomials together.
• Step 2: Out of the two brackets, keep one bracket constant.
• Step 3: Now multiply each and every term from the other bracket.

How Do You Multiply Two Trinomials Together?

Two trinomials can be multiplied together by using the box method as well as the distributive property. Let's take an example to understand the multiplication of two trinomials.

Example: Multiply (5xy + 2x + 3) with (x 2 + 3xy + 7)

• The above given two trinomials are written as (5xy + 2x + 3)× (x 2 + 3xy + 7)
• By using distributive property of polynomial multiplication we get, (5xy + 2x + 3)× (x 2 + 3xy + 7) = 5x 3 y + 15x 2 y 2 + 2x 3 + 6x 2 y + 44xy+ 3x 2 + 14x + 21

Thus, the above multiplication can be shown as (5xy + 2x + 3)× (x 2 + 3xy + 7) = 5x 3 y + 15x 2 y 2 + 2x 3 + 6x 2 y + 44xy + 3x 2 + 14x + 21.