Rules

Before you start to factor any polynomial problem, there are two things you need to check. First, check to make sure the polynomial is in descending order of degree, such as x² + 7x +12.

Second, make sure there is no common monomial that you can factor out of the polynomial. Here is an example:

3x³ + 21x2 + 36

Each term in this polynomial is divisible by 3x, so we remove 3x. Divide each term by 3x. Here is what it will look like:

3x (x² + 7x +12)

Once you have done those two things, you are ready to begin factoring.

Simple Polynomials

Factoring quadratic polynomials is not as hard as it may seem at first. The best way to tackle this concept is to start with an easy one, then work your way to harder, more advanced problems.

Here is a simple polynomial:

x² + 7x + 12

Notice that there is no coefficient in the first term of the polynomial. When there is no coefficient in the first term, factoring begins with listing all of the factors of the last term as such:

Factors of +12

1 x 12, -1 x –12
2 x 6, -2 x –6
3 x 4, -3 x –4

Once you have all of the possible factors, find the pair that adds together to get the coefficient of the second term. In this example, 3 + 4 = 7. These two numbers will be the factors you will use. The answer to this problem is (x + 4) (x + 3). If you check the problem using the FOIL method, you will find that the answer does indeed result in
x2 + 7x + 12.

The same works if you have negative numbers in the equation, such as x2 – 2x – 24. You do the process the same way, by starting with listing the factors of the final term, but note that this time you are listing the factors of a negative number.

Factors of – 24

1 ×–24, -1 x 24
2 x –12, -2 x 12
3 x – 8, -3 x 8
4 x –6, -4 x 6

Again, choose the factors that add up to the coefficient of the second term. Notice that in this example, 4 + -6 = -2. The signs are very important, as the answer to the problem is
(x + 4) (x – 6). If the signs were reversed, the problem would be incorrect.

More Complex Polynomials

When you add a coefficient in front of the first term, the problem becomes more difficult, but certainly not impossible. Here is our example: 3x² +17x +10. Like the previous problems, you start by listing the factors, but this time of both the first and last terms:

Factors of 3

1 x 3

Factors of + 10

1 × 10, -1 ×–10
2 × 5, -2 ×–5 ×

Next, you have to simply start by guessing and checking. Start plugging in factors. Then check by FOIL to see if you get the correct polynomial. Note that you do not have to do the entire FOIL process, but you can skill First and Last terms, and just simply multiply the Outer and Inner parts. Since all of the terms in the example are positive, we only need to concern ourselves with the positive factors. Here is an example:

(1x + 1) (3x + 10) = 3x² + 13x + 10

Notice that the first and last terms are correct but the middle is not. The first and last terms will always be correct, so you can skip finding these, and just find the inner and outer terms. In this example, you would do 1x times 10 and 1 times 3x. This gives you 10x + 3x, which is 13x and not what we want (17x). Before moving to the next set of factors, rearrange these and see if they work:

(1x + 10) (3x + 1)

Find Outer and Inner: 1x + 30x = 31x. Not what we want.

We can now eliminate 1 and 10 as possible solutions, since we tried them in every possible order. Now move to the next factors of 10, 2 and 5

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

Find Outer and Inner: 6x + 5x = 11x. Not what we want. Rearrange.

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

Find Outer and Inner: 15x + 2x = 17x. This is what we want. We have found our solution. Check using FOIL and you get 3x2 + 17x +10.

You can use this same process, even if your first term is not prime. The only difference is that you will have more chances to guess and check. The key to finding the answer to complex polynomials is to be persistent. If you have all of the factors listed, and if you try all possible combinations, you will find the answer. Note that with experience, you can start to get a feel for which factors are more likely to work out correctly. This will speed up the process significantly.