# Factoring

How to multiply polynomials; FOIL method
Ex: (x-2)(x+4) = x2 + 2x - 8

Ex: Factor x2 - 4x + 4

Ex: Solve for x: x2 - 4x + 4 = 0

Methods to factor higher degree polynomials.
Ex: Factor x4 - x3 + x2 - 1

### Multiplying Polynomials

When we multiply polynomials, the most important property to remember is the distributive property. If we want to multiply 4 and the term (x + 3), we know that we have to distribute the 4: 4(x+3) = 4x + 12. This is the same principle we follow when we multiply polynomials. We take each entry of our first term and multiply it to the other term, distributing accordingly, and then add up all of the resultant terms.

Example 1: Multiply (x - 2) and (x + 4).

We would like to multiply together two polynomials. To find (x - 2)(x + 4), we distribute each of the entries in the first term, (x - 2), to the second term, (x + 4). So (x - 2)(x + 4) = x(x + 4) + (-2)(x + 4) = x2 + 4x + -2x - 8 = x2 + 2x - 8

Try out the following exercises to get accustomed with multiplying simple polynomials this way.

Exercises:

Find the following products:

1. (2x - 1)(2x - 1) =

2. x(x + 3)(x - 8) =

3. (4x - 3)(3x + 1.25) =

1. (2x - 1)(2x - 1) = 2x(2x - 1) + (-1)(2x - 1) = 4x2 - 2x - 2x + 1 = 4x2 - 4x + 1

2. In this problem, first we have to recognize that the product has 3 terms: x, (x + 3), and (x - 8). To evaluate a product with more than 2 terms, we simply multiply the first two terms, then multiply that by the next term, until we are done.

x(x + 3)(x - 8) = [x(x + 3)](x - 8) = (x2 + 3x)(x - 8)

Note that now we have reduced the product to just two terms. Now we just multiply as if we only had two terms, seperating and distributing:

(x2 + 3x)(x - 8) = x2(x - 8) + 3x(x - 8) = x3 - 8x2 + 3x2 - 24x = x3 - 5x2 - 24x

3. (4x - 3)(3x + 1.25) = 4x(3x + 1.25) - 3(3x + 1.25) = 12x2 +5x - 9x - 3.75 = 12x2 - 4x - 3.75

### FOIL Method

The FOIL Method is a popular pneumonic device for multiplying simple, first-order polynomials. It is not as thorough or applicable as the distributive method described above, but many students find it easy and it is widely taught in high school classes. For that reason, we will include a brief description.

The FOIL Method is really a reminder of what terms to multiply together to get the correct product. It stands for First - Outer - Inner - Last, where each word describes the entries that you multiply together. We can use our previous example, (x - 2)(x + 4), to demonstrate the method. First, we make the four multiplications, and then add them together.

First: We multiply the first entries of the two terms together.
(x - 2)(x + 4): x * x = x2

Outer: We multiply the outer entries (the ones on the sides) together.
(x - 2)(x + 4): x * 4 = 4x

Inner: We multiply the inner entries together.
(x - 2)(x + 4): -2 * x = -2x

Last: We multiply the last entries of each term together.
(x - 2)(x + 4): -2 * 4 = -8

Finally, we add up all the terms. (x - 2)(x + 4) = x2 + 4x - 2x - 8 = x2 + 2x - 8, which corresponds with the answer we got using the distributive method.

Note that both the FOIL method and the distributive method got us the same 4 terms {x2, 4x, -2x, -8}. This is because the FOIL method is essentially the same as the distributive method. It just distributes for you - each term is multiplied against each term. This is it is really only necessary to learn the distributive method, especially because it has applications to higher order polynomials.

Exercises:

Try to complete the previous exercises using the FOIL method.

### Multiplying higher-order polynomials

When we multiply polynomials that have more than two entries, things get a little more complicated. However, all we need to remember is the distributive property. Each entry from the first term is multiplied by each entry from the second term.

Example 2: Multiply (x2 + 4x - 1) and (x3 - 2x2 + 4)

We simply need to seperate and distribute.
(x2 + 4x - 1)(x3 - 2x2 + 4) = x2(x3 - 2x2 + 4) + 4x(x3 - 2x2 + 4) - 1(x3 - 2x2 + 4)
(x2 + 4x - 1)(x3 - 2x2 + 4) = x5 - 2x4 + 4x2 + 4x4 - 8x3 + 16x - x3 + 2x2 - 4

At this point, we combine like terms:
(x2 + 4x - 1)(x3 - 2x2 + 4) = x5 - 2x4 + 4x2 + 4x4 - 8x3 + 16x - x3 + 2x2 - 4
(x2 + 4x - 1)(x3 - 2x2 + 4) = x5 + 2x4 - 9x3 + 6x2 + 16x - 4