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Multiplying Polynomials

Multiplying Polynomials with Two or More Terms

We can use the Distributive Property to multiply two polynomials where at least one of the polynomials has two or more terms.

For example, suppose we wish to find the product (a + b)(c + d + e).

Distribute (c + d + e) to both a and b.	(a + b)(c + d + e)	= a(c + d + e) + b(c + d + e)
Distribute both a and b to each term in (c + d + e).		= ac + ad + ae + bc + bd + be

Notice that each term in the first polynomial is multiplied by each term in the second polynomial. That is, a is multiplied by c, d, and e. Also, b is multiplied by c, d, and e.

This suggests a general procedure for multiplying any two polynomials.

Procedure â€” Multiplying Polynomials with Two or More Terms

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

Step 2 Simplify.

Example

Find: (3x - 4)(x³ - 5x + 8)

Solution

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

Multiply 3x by x³, -5x, and 8. Then multiply -4 by x³, -5x, and 8.

(3x - 4)(x³- 5x + 8)

(3x)(x³)

3x⁴

(3x)(-5x)

15x²

(3x)(8)

24x

(-4)(x³)

4x³

(-4)(-5x)

20x

(-4)(8)

Step 2 Simplify

Combine like terms and write the terms in descending order.

= 3x⁴ - 15x² + 24x - 4x³ + 20x - 32

So, (3x - 4)(x³ - 5x + 8) = 3x⁴ - 15x² + 24x - 4x³ + 20x - 32.