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# Multiplying Polynomials

## FOIL Method

From

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)

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

-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)

1. First. Outer. Inner. Last.
x * x + x * 4 + 5 * x + 5 * 4
2. Multiply.
x
2 + 4x + 5x + 20
3. Simplify.
x2 + 9x + 20

### Practice with Negatives

Simplify the following polynomials.

(x - 5)(x - 4)

1. Before you start FOILing, change the negative signs:
(x + -5)(+ -4)
2. Now FOIL: First. Outer. Inner. Last.
x * xx * -4 + -5 * x + -5 * -4
3. Multiply.
x
2 + -4x + -5x + 20
4. Simplify.
x
2 + -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)=