Question

The polynomial p(x)=2x^3-x^2-25x-12p(x)=2x 3 −x 2 −25x−12p, left parenthesis, x, right parenthesis, equals, 2, x, cubed, minus, x, squared, minus, 25, x, minus, 12 has a known factor of (x+3)(x+3)left parenthesis, x, plus, 3, right parenthesis. Rewrite p(x)p(x)p, left parenthesis, x, right parenthesis as a product of linear factors.

Answer 1

p(x) = (2x +1)(x +3)(x -4)

Explanation:

You want to rewrite p(x) = 2x^3 -x^2 -25x -12 as a product of linear factors, given that (x+3) is a known factor.

Quadratic factor

Using synthetic division to divide p(x) by the known factor we find the factorization to be ...

p(x) = (x +3)(2x² -7x -4)

Linear factors

The quadratic is factored by looking for factors of (2·(-4) = -8) that have a sum of -7. Those factors are -8 and +1. This gives the factorization of p(x) as ...

p(x) = (x +3)(2x +1)(2x -8)/2 . . . . . where 2 is the leading coefficient

p(x) = (x +3)(2x +1)(x -4)