Factoring Review
Always Factor The Greatest Common Factor Out First!
Example: To factor 2x3 - 2x2 - 24x, first factor out 2x.
COMMON FORMS
a2 - b2 = (a + b)(a - b)
a4 - b4 = (a2 + b2)(a2 - b2) = (a2 + b2)(a + b)(a - b)
x2 + bx + c = (x + k1)(x + k2) where k1 + k2 = b and k1● k2 = c.
ax2 + bx + c = (a1x + k1)(a2x + k2) where a1● a2=a, k1● k2 = c, and a2k1 + a1k2 = b
Factoring By Grouping
AC + BC + DA + DB = C(A + B) + D(A + B) = (C + D)(A + B)
Example: 2x2 + 9x + 10 = 2x2 + 4x + 5x + 10 = 2x(x + 2) + 5(x + 2) = (x + 2)(2x + 5)
The key here is to find two numbers that add to 9 but multiply out to 2x10 = 20. Those numbers are 4 and 5 so you rewrite 9x as 4x+5x.
There are nearly countless other forms of factorization, but these are the ones most commonly used.
Examples:
Factor 9x2 – 49y4
9x2 – 49y4 =
(3x)2 – (7y2)2 =
(3x + 7y2)(3x – 7y2)
Factor x2 – 2x - 120
We need to find two numbers that add to –2 but multiply to –120. These two numbers will be 10 and –12. The factorization is then
(x + 10)(x + -12) =
(x + 10)(x - 12)
Factor 2x2 + 3x – 14 by listing & checking possibilities resulting in 2x2 and –14.
2x2 + 3x – 14 = the following possibilities that result in 2x2 and –14 when multiplied out. But when we check the multiplication, only (2x + 7)(x – 2) results in
2x2 + 3x – 14.
(2x – 2)(x + 7) = 2x2 + 14x – 2x – 14 = 2x2 + 12x – 14 NO!
(2x – 7)(x + 2) = 2x2 + 4x – 7x – 14 = 2x2 - 3x – 14 NO!
(2x + 2)(x - 7) = 2x2 - 14x + 2x – 14 = 2x2 - 12x – 14 NO!
(2x + 7)(x - 2) = 2x2 - 4x + 7x – 14 = 2x2 + 3x – 14 YES!
Note that there were 4 more possible combinations that did not work involving 14 and 1 as second terms like (2x – 1)(x + 14). Often you will find the correct combination by only trying a small fraction of the total possible combinations.
Factor 2x2 – 2x - 84
First, factor out a greatest common factor of 2 to rewrite 2x2 – 2x – 84 as
2(x2 – x – 42) =
2(x – 7)(x + 6)
Factor 2x2 – 3y - 6x + xy by Grouping
First form 2 groups, each containing a common factor. Also, change all subtractions into the addition of negatives. We get
2x2 + xy + -3y + -6x
Now, factor out common factors.
x(2x + y) + -3(y + 2x) which is the same as x(2x + y) – 3(2x + y) and now factor out (2x + y) to get
(2x + y)(x – 3)