Distribute 5
(
−
9
x
2
−
2
x
)
.
5(−9x
2
−2x).

11. Factoring and solving equations

- A. Factor-

1. Factor 3x2 + 6x if possible.

Look for monomial (single-term) factors first; 3 is a factor of both 3x2

and 6x and so is x . Factor them out to get

3x2 + 6x = 3(x2 + 2x1 = 3x(x+ 2) .

2. Factor x2 + x - 6 if possible.

Here we have no common monomial factors. To get the x2 term

we'll have the form (x +-)(x +-) . Since

(x+A)(x+B) = x2 + (A+B)x + AB ,

we need two numbers A and B whose sum is 1 and whose product is

-6 . Integer possibilities that will give a product of -6 are

-6 and 1, 6 and -1, -3 and 2, 3 and -2.

The only pair whose sum is 1 is (3 and -2) , so the factorization is

x2 + x - 6 = (x+3)(x-2) .

3. Factor 4x2 - 3x - 10 if possible.

Because of the 4x2 term the factored form wli be either

(4x+A)(x +B) or (2x+A)(2x+B) . Because of the -10 the integer possi-

bilities for the pair A, B are

10 and -1 , -10 and 1 , 5 and -2 . -5 and 2 , plus each of

these in reversed order.

Check the various possibilities by trial and error. It may help to write

out the expansions

(4x + A)(x+ B) = 4x2 + (4B+A)x + A8

1 trying to get -3 here

(2x+A)(2x+B) = 4x2 + (2B+ 2A)x + AB

Trial and error gives the factorization 4x2 - 3x - 10 - (4x+5)(x- 2) .

4. Difference of two squares. Since (A + B)(A - B) = - B~ , any

expression of the form A' - B' can be factored. Note that A and B

might be anything at all.

Examples: 9x2 - 16 = (3x1' - 4' = (3x +4)(3x - 4)

x2 - 29 = x2 - (my)* = (x+ JTy)(x- my)

For any of the above examples one could also use the