How to Factor Polynomials Easily

Factoring a polynomial is dividing a polynomial by another polynomial until all the factors are found. This is done algebraically by a number of methods: Dividing out obvious factors, looking for a difference of squares (such as x^2 - 4), or grouping terms together in such a way that terms cancel.

Things You'll Need

• Pencil or pen
• Paper

Instructions

1. Look for obvious factors that are common in all the terms, whether in the numerator or denominator of a fraction. Take the example (12x^6 + 8x^5 - 12x^2 - 8x) / (x - 1). You can see immediately that each term on the top of the fraction can be divided evenly by 4 and x.

   Performing that division results in the fraction [4x (3x^5 + 2x^4 - 3x - 2)] / (x-1).

2. Rearrange terms with obvious common factors together, and factor them separately. The top of the fraction in the example has the term (3x^5 + 2x^4 - 3x - 2). The term 3x^5 inside that can be divided by 3x, so put them together and factor the two terms, resulting in (4x [3x (x^4 - 1) + 2x^4 - 2]) / (x-1). Repeat this with the two remaining terms, 2x^4 and 2.
   The fraction is now (4x [3x (x^4 - 1) + 2(x^4 - 1)]) / (x-1).

3. Rearrange the terms together that share the same polynomials. (x^4 - 1) is being multiplied by both 3x and 2. By using the reverse of the FOIL or distribution method, you can see that (3x + 2)(x^4 - 1), when distributed, becomes the [3x (x^4 - 1) + 2x^4 - 2] term in the numerator. So substituting in the undistributed term produces the following polynomial fraction:
   [4x (3x + 2)(x^4 - 1)] / (x-1).

4. Expand terms that are a difference of squares. The top of the fraction has (x^4 - 1) in it. x^4 is (x^2)^2 and one is its own square. It is useful to know that (x^2 - a^2) = (x + a)(x - a). For the example, both (x^4 - 1) terms can be written (x^2 - 1)(x^2 + 1). Notice also that (x^2 - 1) is another difference of squares that can be written as (x + 1)(x - 1).
   The entire expression becomes [4x (3x + 2)(x^2 + 1)(x + 1)(x - 1)] / (x-1)

5. Cancel polynomials in a fraction that are shared by all the terms. Because the numerator is now a string of multiplied polynomials that can be no further divided, you can cancel the (x-1) on the top with the one on the bottom. The final answer is the following:

   4x (3x + 2)(x^2 + 1)(x + 1)

6. Expand a product of binomials if there are any. For example, (x^2 - x - 6) can be expanded as (x + 2)(x - 3). The third term in the expression is negative, so there will always be a plus term and a minus term. If it was positive, than both binomials will either be a difference or a sum. -6 has eight factors: 6, 3, 2, 1, -1, -2, -3, and -6. The second term in the expression has the coefficient of -1, so you must choose two of the factors of -6 that when multiplied together are -6, and when added are -1. The only choice is 2 and -3, so the two binomials are (x + 2)(x - 3). The first term in the expression will always have its exponent halved when splitting into two binomials.

Tips & Warnings

• You may want to write out the steps on paper so the fraction and each term is more visible, as it can be hard to visualize the in-line text with so many parentheses.