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Multiplication

Site: Clare-Gladwin RESD
Course: Michigan Algebra II KHauck
Book: Multiplication
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Date: Thursday, May 16, 2024, 10:29 PM

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Table of contents

Introduction

When multiplying polynomials it is important to remember the Product Rule of Exponents learned in Algebra I.

The Product Rule of Exponents states when multiplying terms with the same base, add the exponents.
For example: xn . xm = x(n+m)

To multiply terms that have coefficients and one or more variables, multiply the coefficients and apply the product rule of exponents on each variable.

Example Multiply. (2x2y3)(3x2y)

Step 1. Group coefficients and like bases.

2 . 3 . x2 . x2 . y3 . y

Step 2. Multiply coefficients & apply the product rule of exponents.

(2 . 3) x(2+2) y(3+1)

Step 3. Simplify.

6x4y4

Monomial by a Monomial

We begin this section by multiplying a monomial by a monomial. As you saw in the introduction, when working with monomials multiply the coefficients separately and then apply the exponent rules to each variable. Let’s try some examples.

Example 1 Multiply.
(3xy5)(-6x4y2)

Step 1. Group coefficients and like bases.

3 . -6 . x . x4 . y5 . y2

Step 2. Multiply coefficients & apply the product rule of exponents.

(3 . -6) x(1+4) y(5+2)

Step 3. Simplify.

-18x5y7


Example 2 Multiply.
(-12a2b3c4)(-3a2b2)

Step 1. Group coefficients and like bases.
-12 . -3 . a2 . a2 . b3 . b2 . c4

Step 2. Multiply coefficients & apply the product rule of exponents.
(-12 . -3) a(2+2) b(3+2)(c4)

Step 3. Simplify.
36a4b5c4

Polynomial by a Monomial

To multiply a polynomial by a monomial, we use the Distributive Property, which says: a(b + c) = ab + ac. Replacing the variables with whole numbers demonstrates how the distributive property works. Using a = 2, b = 3, and c = 4;

Apply the Order of Operations

Apply the Distributive Property

2 (3 + 4) =

2 (3 + 4) =

2 ? 7 =

2 ? 3 + 2 ? 4 =

14

6 + 8 =


14

Since both methods produce the same solution, this demonstrates that the Distributive Property works.

Examples

Example 1 Multiply & write in simplest form.

4x(3x2 - 7)

Step 1. Apply the Distributive Property

(4x)(3x2) + (4x)(-7)


Step 2. Combine like terms if needed & write in standard form.

12x3 - 28x

Example 2 Multiply & write in simplest form.

-7y(4y2 - 2y + 1)

Step 1. Apply the Distributive Property


(-7y)(4y2) + (-7y)(-2y) + (-7y)(1)


Step 2. Combine like terms if needed & write in standard form.

-28y3 + 14y2 - 7y

Example 3 Multiply & write in simplest form.

2x3(3x4 + 2x3 - 10x2 + 7x + 9)

Step 1. Apply the Distributive Property

(2x3)(3x4) + (2x3)(2x3) + (2x3) (-10x2) + (2x3)(7x) + (2x3)(9)


Step 2. Combine like terms if needed & write in standard form.

6x7 + 4x6 - 20x5 + 14x4 + 18x3


Polynomial by a Polynomial

Let’s start by multiplying two binomials, a polynomial with two terms. The Distributive Property also applies in this situation.

Example Multiply & simplify.

(2x + 3)(4x2 + 5)

Step 1. Multiply each term in the first binomial through the second binomial.

2x(4x2 + 5) + 3(4x2 + 5)

Step 2. Continue to use the Distributive Property on the two new terms.

(2x)(4x2) + (2x)(5) + (3)(4x2) + (3)(5)

Step 3. Multiply the monomials in each term.

8x3 + 10x + 12x2 + 15

Step4. Combine like terms if needed and write in standard form.

8x3 + 12x2 + 10x + 15

 

What you should notice is that when multiplying any two polynomials, every term in one polynomial is multiplied by every term in the other polynomial.



Examples

Example 1 Multiply & simplify.

(4x - 5)(x - 20)

Step 1. Multiply each term in the first binomial through the second binomial.

(4x)(x - 20) + (-5)(x -20)

Step 2. Continue to use the Distributive Property on the two new terms.

(4x)(x) + (4x)(-20) + (-5)(x) + (-5)(-20)

Step 3. Multiply the monomials in each term.

4x2 - 80x - 5x + 100

Step 4. Combine like terms if needed and write in standard form.

4x2 - 85x + 10

Example 2 Multiply & simplify.

(3x2 + 2x - 5)(2x - 3)

Step 1. Multiply each term in the first polynomial through each term in the second.

(3x2)(2x) + (3x2)(-3) + (2x)(2x) + (2x)(-3) + (-5)(2x) + (-5)(-3)

Step 2. Multiply the monomials in each term.

6x3 - 9x2 + 4x2 - 6x - 10x + 15

Step 4. Combine like terms if needed and write in standard form.

6x3 - 5x2 - 16x + 15

Video Lesson

Multiplying Polynomials

Special Products

We saw that when we multiply two binomials we need to make sure that each term in the first binomial multiplies with each term in the second binomial. Let’s look at another example.

Multiply two linear binomials: (2x + 3)(x + 4). We obtain 2x2 + 8x + 3x +12, a quadratic polynomial with four terms. The middle terms are like terms and we can combine them. We simplify and get: 2x2 + 11x + 12. The product is a quadratic or 2nd degree trinomial, a polynomial with three terms.

Every time two linear binomials with one variable are multiplied, the result is a quadratic polynomial. This section will introduce special products of binomials.


Square of a Binomial

A special binomial product is the square of a binomial. Consider the following multiplication.

(x + 4)2 is the same as (x + 4)(x + 4)= x2 + 4x + 4x + 16
= x2 + 8x +16.

Notice that the middle terms are the same. Is this a coincidence? Here is another example.

(a – b)2 = (a – b)(a – b) = a2 – ab – ab + b2
= a2 – 2ab + b2.

It looks like the middle terms are the same again. We notice a pattern when squaring binomials. To square a binomial:

  • Square the first term
  • Add twice the product of both terms
  • Add the square of the second term

Example

(x + 10)2= (x)2 + 2(10)(x) + 102 = x2 + 20x + 100

Formulas to Remember

(a + b)2 = a2 + 2ab + b2

and

(a – b)2 = a2 – 2ab + b2


*Note: Don’t make the common mistake of saying (a + b)2 = a2 + b2, they are not equal. To see why, try substituting numbers for a and b into the equation (for example, a = 4 and b = 3) and you will see that it is not a true statement. The middle term, 2ab, is needed to make the equation true.

Examples

Example 1 Square the binomial & simplify.

(5x – 2y)2

Step 1. Apply the Square of a Binomial Rule.

(5x)2 + 2(5x)(-2y) +(-2y)2

Step 2. Simplify & rewrite in standard form.

25x2 – 20xy + 4y2

Example 2 Square the binomial & simplify.

(x2 + 4)2

Step 1. Apply the Square of a Binomial Rule.

(x2)2 + 2(x2)(4) + (4)2

Step 2. Simplify & rewrite in standard form.

x4 + 8x2 + 16

Sum and Difference Patterns

Another special binomial pattern is the product of a sum and a difference of terms. For example,

(x + 4)(x – 4) = x2 – 4x + 4x – 16
= x2 – 16

The middle terms are opposites, and they cancel out when combining like terms. This is not a coincidence. It always happens when multiplying a sum and difference of the same terms.

(a + b)(a – b) = a2 – ab + ab – b2
= a2 – b2

We get the square of the first term minus the square of the second term. You should remember this formula.

Sum and Difference Formula

(a + b)(a – b) = a2 – b2

Examples

Example 1 Multiply the binomials & simplify.

(2x3 + 7)(2x3 - 7)

Step 1. Multiply.

(2x3)2 - 72

Step 2. Simplify & rewrite in standard form.

4x6 - 49

Example 2 Multiply the binomials & simplify.

(4x + 5y)(4x - 5y)

Step 1. Multiply.

(4x)2 - (5y)2

Step 2. Simplify & rewrite in standard form.

16x2 - 25y2

Interactive Activity

To see how to multiply polynomials together select the following link:

Multiply Polynomials

Guided Practice

To solidify your understanding of multiplying polynomial expressions, visit the following link to Holt, Rinehart and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.

Guided Practice

Practice

Multiplying Polynomials Worksheet

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Answer Key

Multiplying Polynomials Answer Key

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Sources

Sources used in this book:

CK12 Foundation, "Factoring Polynomials: More on Probability." 11/6/2009.http://ck12.org/flexr/viewer/962d35c81e3e9e1f4a10fdb462386484/ (accessed 12/3/2009).

Cosmeo, "Multiply Polynomial."2009.http://www.webmath.com/polymult.html (accessed 6/4/2010).

Embracing Mathematics, Assessment & Technology in High Schools; a Michigan Mathematics & Science Partnership Grant Project

Holt, Rinehart and Winston, "Exponents and Polynomials Homework Help Online." http://my.hrw.com/math06_07/nsmedia/homework_help/alg1/alg1_ch07_07_homeworkhelp.html (accessed 6/16/2010).