List of Notes - Category: Factoring Polynomials

Factoring Common Factor

Factoring is the process of rewriting polynomial into product of factors.

Factoring polynomials is quite similar to factoring numbers, but harder (because you need to work with variables) and not exactly the same.

Factoring by Grouping and Regrouping

Factoring by Grouping and Regrouping (factoring in "pairs") is a method for factoring polynomials, that can be applied sometimes, when terms don't have common factors.

Note, that this technique is not always applicable, because there are cases, when it is very hard (or even impossible) to see possibilities for factoring.

Factoring Quadratics

Factoring Quadratics (polynomial of second degree) can be done using factoring by grouping and regrouping (actually, we already saw such example in that note).

Let's try to solve some examples.

Example 1. Solve `x^2+7x+10`.

Difference of Squares

Difference of squares (something squared minus something else squared):

` huge color(purple)(a^2-b^2=(a-b)(a+b)) `

Proof of this fact is straightforward.

We just prove it from right to left.

Apply FOIL to `(a-b)(a+b)`:

Sum and Difference of Cubes

Sum and Difference of Cubes:

$$$\huge{\color{purple}{a^3 \pm b^3=\left(a\pm b\right)\left(a^2 \mp ab+b^2 \right)}}$$$

Proof of this fact is straightforward.

We just prove it from right to left.

Using Techniques for Factoring Together

Now, it is time to understand how to apply learned techniques together.

Recall, that we've learned following factoring techniques:

  • Factoring Common Factor
  • Factoring by Grouping and Regrouping
  • Factoring Quadratics
  • Difference of Squares
  • Sum and Difference of Cubes

To be successful in factoring polynomials, you need to recognize when and what method to use.