Topics
9. Polynomials and Factoring
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United States of America
Grades 9-12

9.02 Multiplying polynomials

Lesson
Practice

Introduction

Just as we learned about polynomial addition in lesson  4.01 Adding and subtracting polynomials  , multiplying polynomials will also result in a polynomial. We will connect the multiplication of integers to polynomial multiplication and practice various examples.

Multiplying polynomials

To multiply two polynomials together, we make use of the distributive property:a\left(b+c\right)=ab + ac

Recall that ax^m \cdot bx^n = abx^{m+n}. Consider the expanded form of the product property of exponents:

\displaystyle 5x^3 \cdot 4x^2\displaystyle =\displaystyle 5 \cdot x \cdot x \cdot x \cdot 4 \cdot x \cdot x
\displaystyle =\displaystyle 5 \cdot 4 \cdot x \cdot x \cdot x \cdot x \cdot x
\displaystyle =\displaystyle 20x^5

Exploration

Complete the area models for multiplication shown:

A rectangle divided into 2 rectangles. The side next to the left rectangle is labeled 3 while the top is labeled 7. The top of the right rectangle is labeled 9.
3(7+9)
A rectangle divided into 2 rectangles. The side next to the left rectangle is labeled 3 x squared while the top is labeled 7 x cubed. The top of the right rectangle is labeled 9.
3x^2(7x^3+9)

Complete the new area models for multiplication:

A rectangle divided into 2 rows of 2 rectangles. The outside of the rectangle is labeled 20 above the left column and 1 above the right column. The side is labeled 10 next to the top row and negative 3 next to the bottom row.
(10-3)(20+1)
A rectangle divided into 2 rows of 2 rectangles. The outside of the rectangle is labeled 2 x above the left column, 1 above the right column, and a plus sign above the line between the left and right column. The side is labeled x next to the top row and negative 3 next to the bottom row.
(x-3)(2x+1)

  1. What do the area models have in common?

  2. What's different about the area models?
  3. Make a conjecture about how multiplying polynomials relates to multiplying integers.

Area models can help us organize the multiplication of polynomaials, so we don't forget to distribute any terms. Then, we can combine like terms to get the simplest polynomial.

When we multiply two polynomials, the product is a polynomial. Similarly, when we multiply two integers, the product is an integer.

The product of two polynomials will always result in a new polynomial where

  • The degree of the new polynomial will be the sum of the degrees of the multiplied polynomials.
  • The number of terms may vary from the original polynomials depending on how like terms are combined.

Examples

Example 1

Multiply 3 x \left( 2 x^{2} - 5 x + 4\right).

Worked Solution
Apply the idea

We can use the distributive property to get the product of the monomial 3 x and the trinomial 2 x^{2} - 5 x + 4.

\displaystyle 3 x \left( 2 x^{2} - 5 x + 4\right)\displaystyle =\displaystyle 6 x^{3} - 15 x^{2} + 12x

Since there are no more like terms and the expression is already in standard form, the final answer is 6 x^{3} - 15 x^{2} + 12x.

Reflect and check

6 x^{3} - 15 x^{2} + 12x is considered a polynomial of degree 3, since 3 is the value of the highest exponent on a variable in the polynomial.

Example 2

Consider the polynomials 7 y + 2 and 4 y - 5.

a

Find the product of the two polynomials.

Worked Solution
Create a strategy

Multiply the polynomials using distribution.

Apply the idea

We can use the distributive property to get the product of the two binomials 7y + 2 and 4 y - 5.

\displaystyle \left( 7 y + 2\right) \left( 4 y - 5\right)\displaystyle =\displaystyle 4y\left( 7 y + 2\right) - 5\left( 7 y + 2\right)Distributive property
\displaystyle =\displaystyle 28 y^{2} + 8y - 35 y - 10Distributive property
\displaystyle =\displaystyle 28y^{2} - 27y - 10Combine like terms

Since the expression is already in standard form, the final answer is 28y^{2} - 27y - 10.

Reflect and check

We can also use an area model to multiply the two polynomials:

A rectangle divided into 2 rows of 2 rectangles. The top left rectangle is labeled 28 y squared, the bottom left rectangle is labeled negative 35 y, the top right rectangle is labeled 8 y and the bottom right rectangle is labeled negative 10. The side of the rectangle is labeled 4 y next to the top right rectangle, and negative 5 next to the bottom right. The top of the rectangle is labeled 7 y + 2.
b

Explain why the product of two polynomials is also a polynomial.

Worked Solution
Apply the idea

A polynomial is a collection of terms in the form mx^n where m is a real number and n is a non-negative integer.

We know that product of two algebraic terms with non-negative integer exponents together results in an algebraic term with non-negative integer exponents. Since multiplying polynomials together results a sum of such products, by definition the result is a polynomial expression.

Reflect and check

We can use this explanation to think about what happens when we perform multiple operations on polynomials. What happens if we add two polynomials and multiply this result by another polynomial? What if we multiply three polynomials? Is our result still a polynomial?

Example 3

Inhar is designing a cubic storage container with odd-numbered side lengths. They decide to let 2x+1 yards represent the length of each side.

a

Confirm that the side length will always be odd.

Worked Solution
Create a strategy

Analyze the given side length.

Apply the idea

Since 2x+1 represents the side lengths of the container, we can note that twice any number is always an even product. If we add an odd number like 1 to that, the side length 2x+1 will always be odd.

b

Write an expression for the surface area of the storage container.

Worked Solution
Create a strategy

Draw and label a diagram of the storage container first, then use it to calculate the surface area.

Apply the idea
A cube. The front view of the cube is divided into 2 rows of 2 squares. The side next to the top left square is labeled 2 x and the side next to the bottom left square is labeled 1. The bottom of the bottom left square is labeled 2 x and the bottom of the bottom right square is labeled 1.

We can calculate the area of one face on the storage container, then multiply the polynomial expression by 6 faces on the cube-shaped container.

A cube. The front view of the cube is divided into 2 rows of 2 squares. The top left square is labeled 4 x squared, the bottom left square is 2 x, the top right square is 2 x and the bottom right is 1. The side next to the top left square is labeled 2 x and the side next to the bottom left square is labeled 1. The bottom of the bottom left square is labeled 2 x and the bottom of the bottom right square is labeled 1.

Area of one face: 4x^2+2x+2x+1=4x^2+4x+1 square yards

Surface area of container: 6(4x^2+4x+1)=24x^2+24x+6 square yards

c

Write an expression for the volume of the storage container.

Worked Solution
Create a strategy

Use the formula for the volume of a cube to calculate the volume of the storage container.

Apply the idea

Since the formula for the volume of a cube is V=l \cdot w \cdot h, we can calculate the volume of the storage container as shown:

\displaystyle V\displaystyle =\displaystyle (2x+1)(2x+1)(2x+1)Substitute expressions for l,w, and h
\displaystyle =\displaystyle (2x+1)[(4x^2+2x+2x+1)]Distributive property
\displaystyle =\displaystyle (2x+1)(4x^2+4x+1)Combine like terms
\displaystyle =\displaystyle (8x^3+8x^2+2x)+(4x^2+4x+1)Distributive property
\displaystyle =\displaystyle 8x^3+12x^2+6x+1 \text{ cubic yards}Combine like terms
Reflect and check

A labeled diagram of the storage container can help us conceptualize the problem.

A cube with labels: Height: quantity 2 x + 1 yards for the height, Length: quantity 2 x + 1 yards for the length and Width: quantity 2 x + 1 yards for the width.
Idea summary

Polynomials can be multiplied using the distributive property. Using an area model for multiplying polynomials helps keep track of terms.

Special products of binomials

For some products of binomials, we can look for patterns to help us simplify more efficiently.

Exploration

Consider the expansions of the following binomials of the form (a+b)(a+b)=(a+b)^2:

  • (x+3)(x+3) = x^2 + 6x + 9
  • (x+5)(x+5)= x^2 + 10x+ 25
  • (x+6)^{2}=x^2 + 12x + 36
  • (2r+3s)^{2}=4r^{2} + 12rs + 9s^{2}
  • (5s-3)^{2}= 25s^{2} - 30s + 9
  1. What do you notice about the linear coefficient of the product?
  2. What do you notice about the constant of the product?
  3. Is there a general rule for this type of product?

Consider the expansions of the following binomials of the form (a+b)(a-b):

  • (x+3)(x-3)=x^2-9
  • (x-5)(x+5)=x^5-25
  • (x+6)(x-6)=x^2-36
  • (2r+3s)(2r-3s)=4r^{2}-9s^{2}
  • (5s-3)(5s+3)=25s^{2}-9
  1. What do you notice about the linear coefficient of the product?
  2. What do you notice about the constant of the product?
  3. Is there a general rule for this type of product?

In general,

  • (a+b)(a+b)= (a+b)^2=a^{2}+2ab + b^{2}

  • \left(a+b\right)\left(a-b\right) = a^{2} - b^{2}

This leads to special products which are special cases of products of polynomials. With special products, we can multiply two polynomials without using the distributive property.

For binomials, we have the following special binomial products, which are called identities:

Square of a binomial (Sum)

\left(a + b\right)^{2} = a^{2} + 2 a b + b^{2}

Square of a binomial (Difference)

\left(a - b\right)^{2} = a^{2} - 2 a b + b^{2}

Product of a sum and difference

\left(a+b\right)\left(a-b\right) = a^{2} - b^{2}

This is the identity of the difference of two squares.

Note: \left(a + b\right)^{2} \neq a^{2} + b^{2} and \left(a - b\right)^{2} \neq a^{2} - b^{2}.

Examples

Example 4

Multiply and simplify the following binomials.

a

\left(x - 4\right)^{2}

Worked Solution
Create a strategy

We check first whether \left(x - 4\right)^{2} is a special binomial product and identify its form.

Apply the idea

Since \left(x - 4\right)^{2} is a square of a binomial in the form \left(a - b\right)^{2}, we use the formula and simplify the expression:

\displaystyle \left(a - b\right)^{2}\displaystyle =\displaystyle a^{2} - 2 a b + b^{2}Identity for the square of a binomial
\displaystyle \left(x - 4\right)^{2}\displaystyle =\displaystyle x^{2} - 2\left(x\right)\left(4\right) + 4^{2}Substitute a=x and b=4
\displaystyle =\displaystyle x^{2} - 8 x + 16Evaluate the multiplication and exponent
b

\left(x + 4\right)\left(x - 4\right)

Worked Solution
Create a strategy

We check first whether \left(x + 4\right)\left(x - 4\right) is a special binomial product and identify its form.

Apply the idea

Since \left(x + 4\right)\left(x - 4\right) is a product of a sum and difference, we use the formula and simplify the expression:

\displaystyle \left(a+b\right)\left(a-b\right)\displaystyle =\displaystyle a^{2} - b^{2}Identity for the product of a sum and difference
\displaystyle \left(x+4\right)\left(x-4\right)\displaystyle =\displaystyle x^{2} - 4^{2}Substitute a=x and b=4
\displaystyle =\displaystyle x^{2} - 16Evaluate the exponent
c

\left(2x + 5\right)\left(2x - 5\right)

Worked Solution
Create a strategy

We check first whether \left(2x + 5\right)\left(2x - 5\right) is a special binomial product and identify its form.

Apply the idea

Since \left(2x + 5\right)\left(2x - 5\right) is a product of a sum and difference, we use the formula and simplify the expression:

\displaystyle \left(a+b\right)\left(a-b\right)\displaystyle =\displaystyle a^{2} - b^{2}Identity for the product of a sum and difference
\displaystyle \left(2x+5\right)\left(2x-5\right)\displaystyle =\displaystyle \left(2x\right)^{2} - 5^{2}Substitute a=2x and b=5
\displaystyle =\displaystyle 4x^{2} - 25Evaluate the exponents
d

Multiply and simplify: 3\left(2x + 5y\right)^{2}

Worked Solution
Create a strategy

We check first whether 3\left(2x + 5y\right)^{2} involves a special binomial product and identify its form.

Apply the idea

Since \left(2x + 5y\right)^{2} is a square of a binomial in the form \left(a + b\right)^{2}, we use the formula and simplify the expression:

\displaystyle \left(a + b\right)^{2}\displaystyle =\displaystyle a^{2} + 2 a b + b^{2}Identity for the square of a binomial
\displaystyle 3 \left(2x + 5y\right)^{2}\displaystyle =\displaystyle 3 \left[\left(2x\right)^{2} + 2\left(2x\right)\left(5y\right) + \left(5y\right)^{2}\right]Substitute a=2x and b=5y and multiply by 3
\displaystyle =\displaystyle 3 \left[4x^{2} + 20xy + 25y^{2}\right]Evaluate the exponents and multiplication
\displaystyle =\displaystyle 12x^{2} + 60xy + 75y^{2}Distributive property
Idea summary

Recognizing the patterns in special binomial factors may be helpful in multiplication problems and upcoming lessons. Remember the patterns:

  • Square of a binomial (Sum): \left(a + b\right)^{2} = a^{2} + 2 a b + b^{2}
  • Square of a binomial (Difference): \left(a - b\right)^{2} = a^{2} - 2 a b + b^{2}
  • Product of a sum and difference: \left(a+b\right)\left(a-b\right) = a^{2} - b^{2}

Outcomes

A.APR.A.1

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.

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