FOIL

This is a pretty easy topic...seeing that we already know the distributive property...let's refresh ourselves on that:

Distributive Property: the number outside the parentheses is multiplied by every number within the parentheses

Ex. 3(4 + 5) = 3(4) + 3(5) = 12 + 15 = 27

2(6 - 3) = 2(6) - 2(3) = 12 - 6 = 6

3(x + 4 + 3) = 3(x) + 3(4) + 3(3) = 3x + 12 + 9 = 3x + 21

x(x + 2) = x2 + 2x

You remember this, right?  You are just combining like terms.  The reason we use it is because of variables (like in the last two examples).  We may need to simplify something without knowing the values of all the numbers.  However, we have just been working with one number times many numbers...what about many numbers times many numbers...

First here's a definition...

Polynomial: numbers or variables with nonnegative integer powers of the variable

Ex. 2x, 3 + 7y, 5x3, 20y2 - 9

Ex. (not polynomials). 1/x + 4 (remember 1/x = x-1), √y (this is equal to y1/2), 2x1/4

Ok...now there are certain different types of polynomials...

Monomial: one term in the polynomial (4, 5x, 3y3)

Binomial: two terms in the polynomial (5 + 2x, x2 - 4x)

Trinomial: three terms in the polynomial (you get this)

Polynomials are also recognized by their degrees.

Degree: exponent of the highest power term of the polynomial

Ex. A monomial of degree 4: 5x4

We should also put polynomials in standard form once they're simplified.

Standard Form: simplified as much as possible and the powers of variables decrease from left to right

Now on to FOIL.

Let's say we want to multiply (3x + 2)(4x - 3).
FOIL stands for something:
First
Outer
Inner
Last

So, step 1 is to multiply the first terms of the binomials:
3x * 4x = 12x2
Step 2 is to multiply the outer terms of the binomials (looking at them):
3x * -3 = -9x
Step 3 is to multiply the inner terms of binomial:
2 * 4x = 8x
Step 4, the last one, is to multiply the last terms of the binomials:
2 * -3 = -6
Now we have 12x2 - 9x + 8x - 6
Combine like terms and get: 12x2 - x - 6

Basically what we do is multiply each term in the first binomial by each one in the second.  The FOIL method only works for binomials, but this principle behind it works for all polynomials.  Let's try another one...
(x + 2)(x - 4)
x(x) = x2, x(-4) = -4x, 2(x) = 2x, 2(-4) = -8
x2 - 4x + 2x - 8
x2 - 2x - 8

How about a binomial * a trinomial...
(x + 5)(x2 - 2x + 6)
x(x2) = x3, x(-2x) = -2x2, x(6) = 6x, 5(x2) = 5x2, 5(-2x) = -10x, 5(6) = 30
x3 - 2x2 + 6x + 5x2 - 10x + 30
x3 + 3x2 -4x + 30

Summary of Steps to Multiplying Polynomials
1. Multiply the first term in the first polynomial by each term in the second.  Then do it with the second, the third, etc. (If you have more than 2 polynomials to multiply, multiply two, then multiply that by the third, etc.)
2.  Combine like terms.

Here are a few practice problems...below them are a link to the answers (not step by step).

1. (x + 4)(x - 8)
2. (3x - 3)(x + 2)
3. (2x + 1)2
4. (x + 9)(2x2 - 3x)

5. (x + 3)(x + 2)(x + 1)