Question

What is the completely factored form of f(x) = 6x3 – 13x2 – 4x + 15?

Answer 1

1. Determine the possible roots using the rational root theorem

Using the rational root theorem the possible roots are

\(P = \pm 1, \pm 3, \pm 5 \pm 15\)
\(Q = \pm 1, \pm 2, \pm 3, \pm 6\)
\(\frac{P}{Q} = \pm1\pm3, \pm3, \pm5, \pm15, \pm\frac{3}{2}, \pm\frac{1}{2}, \pm\frac{1}{3}, \pm\frac{1}{6}, \pm\frac{5}{2}, \pm\frac{5}{3}, \pm\frac{15}{2}\)

Answer 2

Now \(f(-1) = 0\) which means \(x= -1\) and \(x +1 = 0\). thus \((x +1)\) is a factor.

Using polynomial division:
\(f(x) = (x + 1)(6x^2 - 19x + 15)\)

Factoring \(6x^2 - 19x + 15\) yields \((3x - 5)(2x - 3)\)

Thus the factored form of the given function is
\(f(x) = (x + 1)(3x-5)(2x -3)\)