Unit 2: Factoring Polynomials

There are several methods to factoring polynomials, and several types of polynomials. Most, however, come as quadratic equations in the form ax²+bx+c.

Factoring quadratic equations with a leading (a) coefficient of 1 is relatively simple. When factored, the polynomial will be expressed as a binomial.
The quadratic involves two operations; starting off, you need to observe the second one. If it is negative, then the factored quadratic will be written as (x + __)(x - __). If it is positive, then you must observe the first operation, which will determine the operation defined in both factors of the factored quadratic, either (x + __)(x + __) or (x - __)(x - __).
After determining how the quadratic will be expressed, you need to find two factors of the third coefficient (c) that add or subtract to the middle coefficient (b), depending on the quadratic.
When you determine these two factors, simply place them in their corresponding places, and you will have successfully factored the quadratic.

Factoring quadratics with a leading coefficient (a) that is not 1 is somewhat more complicated. First off, you need to multiply the leading coefficient with the last coefficient (c) in order to yeild the master product.
Find two factors of the master product that add to the middle coefficient (b), and place these two factors in their corresponding places.
The result will have an extra factor of the leading coefficient (a), so divide it out and simplify. 

If a binomial represents the difference of two squares, then a² – b² = (a + b)(a – b).

A quadratic with two equal factors can be factored in two ways:

• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²

If a quadratic involves higher powers, then z-substitution may be used.
ex. if z = x², then x⁴ – 6x² + 8 = z² – 6z + 8

The sum or differene of 2 cubes can be expressed in two ways:

• a³ + b³ = (a + b)(a² – ab + b²)
• a³ - b³ = (a - b)(a² + ab + b²)