Multiplying Polynomials

We will consider four cases of polynomial multiplication.

Case 1: Multiply two monomials.

Multiply the coefficients. Then use the Multiplication Property of Exponents to combine variable factors that have the same base.

Note:

Multiplication Property of Exponents: x^mxⁿ = x^m+n

 

Example 1

Find: (-5w³y²) · (7wy⁵)

 

Case 2: Multiply a polynomial by a monomial.

Use the Distributive Property to remove the parentheses. Then, for each term, multiply the resulting monomials.

Note:

Recall the Distributive Property: a(b + c) = ab + ac

 

Example 2

Find: -4y²(3y⁴ + 2wy³ - 7)

So, the product is -12y⁶ - 8wy⁵ + 28y².

 

Case 3: Multiply a polynomial by a polynomial.

To find the product of two polynomials, multiply each term of one polynomial by each term of the other polynomial.

 

Example 3

Find: (2x - 7)(5x² - 6x + 9)

Solution

Multiply each term of the trinomial by 2x and by -7.

(2x - 7)(5x² - 6x + 9)

= 2x · 5x² + 2x · (-6x) + 2x · 9 + (-7) · 5x² + (-7) · (-6x) + (-7) · 9

= 10x³ - 12x² + 18x - 35x² + 42x - 63

= 10x³ - 47x² - 60x - 63

So, the product is 10x³ - 47x² - 60x - 63.

 

Case 4: Multiplying two binomials.

The product of two binomials can be represented as follows:

(a + b)(c + d) = ac + ad + bc + bd

You can use the acronym FOIL to help you remember this formula. FOIL stands for First, Outer, Inner, Last.

Note:

This picture may help you remember how to use the FOIL method. The arcs form a "face".

 

Example 4

Find: (5x - 7y)(6x + 9y)

Solution

So, the product is 30x² + 3xy - 63y².