How to factor and write the full equation

Question

Given P(x)=6x^5-17x^4-15x^3+33x^2+13x-12, write P in factored form. Be sure to write the full equation, including P(x)=

Answer 1

Try to find whole roots of this polynomial.

If a whole root exists, it should be a divisor of the constant term (-12).

So, whole roots could be only ±1, ±2, ±3, ±4, ±6, ±12.

Check these values. We could stop when we find that x = -1 is a root.

Divide the original polynomial by (x+1):

(6x⁵ - 17x⁴ - 15x³ +33x² + 13x - 12) ÷ (x + 1) = 6x⁴ - 23x³ + 8x² + 25x - 12

Now try to find the whole roots for the new polynomial (again from ±1, ±2, ±3, ±4, ±6, ±12).

Again, the value x = -1 is a root of the new polynomial.

Divide the new polynomial by (x + 1):

(6x⁴ - 23x³ + 8x² + 25x - 12) ÷ (x + 1) = 6x³ - 29x² + 37x - 12

Try to find the whole roots for this polynomial.

The value x = 3 is a root.

Divide this polynomial by (x - 3):

(6x³ - 29x² + 37x - 12) ÷ (x - 3) = 6x² -11x + 4

The last polynomial is a binomial, and we can factor them by k-l method:

6x² -11x + 4 = (2x - 1)(3x - 4)

Finally, P(x) = (x + 1)² (x - 3) (2x - 1) (3x - 4)

Answer 2

If you're expected to do this by hand, you may have a bit of a sadistic teacher. If you're allowed to use outside resources, head over to https://www.wolframalpha.com/ and type "factor 6x^5-17x^4-15x^3+33x^2+13x-12". The answer it pops out is P(x) = (-3 + x)(1 + x)²(-1 + 2 x)(-4 + 3 x).