__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) = x^{2} + 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, 5x^{3}, 20y^{2} - 9

**Ex. (not polynomials).** 1/x + 4 (remember 1/x = x^{-1}),
√y (this is equal to y^{1/2}), 2x^{1/4}

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

__Monomial:__
one term in the polynomial (4, 5x, 3y^{3})

__Binomial:__
two terms in the polynomial (5 + 2x, x^{2} - 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:
5x^{4}

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:

**F**irst

**O**uter

**I**nner

**L**ast

So, step 1 is to multiply the **first**
terms of the binomials:

3x * 4x = **12x**^{2
}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 =

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) = **x ^{2}**, x(-4) =

x

How about a binomial * a
trinomial...

**(x + 5)(x ^{2} - 2x + 6)**

x(x

x

__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)(2x^{2} - 3x)

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

Click
here for the answers.

Click here to go back
to the index.