Factoring Completely
Some polynomials cannot be factored into the product of two binomials with integer coefficients, (such as x² + 16), and are referred to as prime.
Other polynomials contain a multitude of factors.
“Factoring completely” means to continue factoring until no further factors can be found. More specifically, it means to continue factoring until all factors other than monomial factors are prime factors. You will have to look at the problems very carefully to be sure that you have found all of the possible factors.
To factor completely:
- Search for a greatest common factor. If you find one, factor it out of the polynomial.
- Examine what remains, looking for a trinomial or a binomial which can be factored.
- Express the answer as the product of all of the factors you have found.
Example: Factor completely: 5x2 – 45
- Search for the greatest common factor. In this problem, the greatest common factor is 5.
5x2 – 45 = 5(x2 – 9) - Now, examine the binomial x2 – 9. (Notice how the factor of 5 is tagging along and remains as part of the answer.)
5(x2 – 9) = 5 (x – 3) (x + 3) - Since the binomials (x – 3) and (x + 3) cannot be factored further, we are done. Express the answer as the product of all of the factors.
5 (x – 3) (x + 3)
Example: Factor completely: 4x2 – 24x – 28
- Search for the greatest common factor. In this problem, the greatest common factor is 4.
4x2 – 24x – 28 = 4(x2 – 6x – 7) - Now, examine and factor the trinomial x2 – 6x – 7. Don’t drop the 4.
4(x2 – 6x – 7) = 4 (x – 7) (x + 1) - Since the binomials (x – 7) and (x + 1) cannot be factored further, we are done. Express the answer as the product of all of the factors.
4 (x – 7) (x + 1)