Multiplying Polynomials

After studying this lesson, you will be able to:

• Multiply polynomials.

When multiplying polynomials, we multiply the coefficients and we add the exponents. If we have parentheses, we use the distributive property.

Example 1

5a (3a ² + 4)

We need to distribute 5a. Remember to multiply the coefficients and add the exponents

15a ² + 20a

Example 2

2m ² (5m ² - 7m + 8)

We need to distribute 2m ² . Remember to multiply the coefficients and add the exponents.

10m ⁴ - 14m ³ + 16m ²

Example 3

We need to distribute 2 3 x . Remember to multiply the coefficients and add the exponents.

To multiply by 12 x...we can do 2x times 12x and then divide by 3 to get 8x ²

To multiply by -9...we can do 2x times -9 and then divide by 3 to get -6x.

8x ³ - 6x

Example 4

-3x ² y ( 2x ² y- 3x y ² - 7y ³ )

We need to distribute -3x ² y. Remember to multiply the coefficients and add the exponents.

-6x ⁴ y ² + 9x ³ y ³ + 21x ² y ⁴

Example 5

We need to distribute . Remember to multiply the coefficients and add the exponents. Remember that -y is the same as -1y.

Multiply -y times 20y ² then divide by 2 to get -10y ³.

Multiply -y times -10y then divide by 2 to get 5y ² .

Multiply -y times 4 then divide by 2 to get -2y.

-10y ³ + 5y ² -2y

Example 6

5x ² (x + 7) - 2x (5x ² - 3x + 7) + 2(x ³ - 8)

We have to distribute three times on this problem. Doing so will give us:

5x ³ + 35x ² - 10x ³ + 6x ² - 14x + 2x ³ - 16

Now, we add like terms and put the terms in descending order:

-3x ³ + 41x ² - 14x - 16

Example 7

Solve x(x - 3) + 4x - 3 = 8x + 4 + x (3 + x )

This is an equation. First we need to distribute to remove the parentheses. This will give us:

x ² - 3x + 4x - 3 = 8x + 4 + 3x + x ²

Next, we collect like terms on each side. This will give us:

x ² + x - 3 = 11x + 4 + x ²

Next, we subtract x ² from each side. This will give us:

x -3 = 11x + 4

Now, we subtract x from each side to get: -3 = 10x + 4

Now, we subtract 4 from each side to get: -7 = 10x

Now we divided each side by 10 to get