http://www.ck12.org Chapter 2. Polynomials and Rational Functions
Example A
Factor the following expression: 6x^2 − 13 x− 28
Solution: While this trinomial can be factored by using the quadratic formula or by guessing and checking, it can
also be factored using a factoring algorithm. Here, you will learn how this algorithm works.
6 x^2 − 13 x− 28
First, multiply the first coefficient with the last coefficient:
x^2 − 13 x− 168
Second, factor as you normally would witha=1:
(x− 21 )(x+ 8 )
Third, divide the second half of each binomial by the coefficient that was multiplied originally:
(x− 21
6
)(x+ 8
6)
Fourth, simplify each fraction completely:
(x− 7
2
)(x+ 4
3)
Lastly, move the denominator of each fraction to become the coefficient ofx:
( 2 x− 7 )( 3 x+ 4 )
Note that this is a procedural algorithm that has not been proved in this text. It does work and can be a great time
saving tool.
Example B
Factor the following expression using two methods different from the method used in Example A: 6x^2 − 13 x− 28
Solution: The educated guess and check method can be time consuming, but since there are a finite number of
possibilities, it is still possible to check them all. The 6 can be factored into the following four pairs:
1, 6
2, 3
-1, -6
-2, -3
The -28 can be factored into the following twelve pairs:
1, -28 or -28, 1
-1, 28 or 28, -1
2, -14 or -14, 2
-2, 14 or 14, -2
4, -7 or -7, 4
-4, -7 or -7, -4
The correctly factored expression will need a pair from the top list and a pair from the bottom list. This is 48
possible combinations to try.
If you try the first pair from each list and multiply out you will see that the first and the last coefficients are correct
but thebcoefficient does not.
( 1 x+ 1 )( 6 x− 28 ) = 6 x− 28 x+ 6 x− 28
A systematic approach to every one of the 48 possible combinations is the best way to avoid missing the correct