Factorization

Key points

Factorization of polynomial functions, including quadratics, is a way to algebraically find the zeroes/roots/solutions of the function.
- Done to find the factors of a polynomial function.
To factorize:
- Set the equation equal to 0
- Take out the common factor
- Factor the terms of the polynomial
- Set equal to 0 and solve for x.

Factorization is the process by which the factors of a polynomial (often quadratic) function are found.

Finding the factors and setting them equal to 0 allows you to solve for the roots.
- Factorization is the easiest and quickest way to algebraically solve for the solutions of a quadratic function.
It is essentially the opposite of the FOIL method.
- You are finding the original factors rather than combining them.

Factorization of quadratics consists of a series of steps:

Combine all like terms on one side of the equation so it is equal to 0.
Factor out the greatest common factor.
- Can be done with coefficients and/or with variables, depending on the case.
If the leading coefficient of the quadratic is greater than 1, “split” the middle term as seen in the example.
Find the two binomial terms.
- The factors are in the structure (x+p)(x+q).
  - Within standard quadratic form: p and q add to b, and they multiply to ac.
Set each factor 0 equal to x.
- If the x coefficient is 1, then the solutions will be -p and -q.
Optional Step:
- FOIL to check

Extra Tip:

If the coefficients of the quadratic add up to 0, one of the factors will be (x-1).
- Factor this out from the original quadratic to find the other factor.
  - Set these factors equal to 0 and solve for x