Multiplying Polynomials

Definition of polynomial: A polynomial can also be the difference of 2 or more monomials.

Examples of Polynomials

3x -2
Monomials: 3x, -2

5xy² + 36xy + 52x + 6
Monomials: 5xy², 36xy, 52x, 6

Πr² + 2Πh
Monomials: Πr², 2Πh

When to Multiply Polynomials

The instructions will ask you to multiply or simplify exercises that look like this:

• (Polynomial) × (Polynomial)
• (Polynomial) • (Polynomial)
• (Polynomial) * (Polynomial)
• (Polynomial)^x
• (Polynomial)(Polynomial)

Note:  When there is no multiplication symbol between 2 sets of parentheses, realize that you are being called to multiply.

When to Not Multiply Polymonials

(Polynomial) + (Polynomial)

(Polynomial)  − (Polynomial)

Yes, I understand that parentheses encompass the polynomials, but pay attention to what the exercise is asking you to do.

 (3x + 5y) + (2x +-6y) does not equal (3x + 5y) (2x +-6y).

Practice with Constants

Multiply:  (8 + 6)(-2 + 5)

Use order of operations:

1. Parentheses
   (8+ 6) = 14
   (-2 + 5) = 3
2. Multiply
   14 * 3 = 42

Introducing FOIL

Here’s another way of looking at it:

FOIL is method to multiply polynomials. It is an acronym for First, Outer, Inner, Last

1. First: (8 + 6)(-2 + 5)
Multiply the first terms: 8 * -2 = -16

2. Outer: (8 + 6)(-2 + 5)
Multiply the outer terms: 8 * 5 = 40

3. Inner: (8 + 6)(-2 + 5)
Multiply the inner terms: 6 * -2= -12

4. Last: (8 + 6)(-2 + 5)
Multiply the outer terms: 6 * 5 = 30

Next, add the results:

-16 + 40 + -12 + 30

Simplify:

-16 + 40 + -12 + 30 = 42

Now, let's practice multiplying polynomials with variables.

Practice with Positives

Simplify the following polynomials.

(x + 5)(x + 4)

First. Outer. Inner. Last.
x * x + x * 4 + 5 * x + 5 * 4

Multiply.
x² + 4x + 5x + 20

Simplify.
x² + 9x + 20

Practice with Negatives

Simplify the following polynomials.

(x - 5)(x - 4)

Before you start FOILing, change the negative signs:
(x + -5)(x + -4)

Now FOIL: First. Outer. Inner. Last.
x * x + x * -4 + -5 * x + -5 * -4

Multiply.
x² + -4x + -5x + 20

Simplify.
x² + -9x + 20 

Practice Exercises

1) (x + 3)(x - 3) =

2) (x - 6)(x + 4) =

3) (x - 8)(x - 9) =

4) (5j + 11)(j + 1)=

5) (5p - 7)(4p + 3)=

Answers & Explanations