Question

Factor the following complex polynomials using the decomposition method: show all work.
a. 15x^2 + x - 6

Answer 1

Factoring 15x2-x-6

The first term is, 15x2 its coefficient is 15.
The middle term is, -x its coefficient is -1.
The last term, "the constant", is -6


Step-1: Multiply the coefficient of the first term by the constant 15 • -6 = -90

Step-2: Find two factors of -90 whose sum equals the coefficient of the middle term, which is -1.

-90 + 1 = -89
-45 + 2 = -43
-30 + 3 = -27
-18 + 5 = -13
-15 + 6 = -9
-10 + 9 = -1 That's it


Step-3: Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -10 and 9
15x2 - 10x + 9x - 6

Step-4 : Add up the first 2 terms, pulling out like factors :
5x • (3x-2)
Add up the last 2 terms, pulling out common factors :
3 • (3x-2)

Step-5: Add up the four terms of step 4 :
(5x+3) • (3x-2)
Which is the desired factorization

Final Answer - (3x - 2) • (5x + 3)