How should I start Factoring this?

Question

So im supposted to factor this and I'm not sure where to start, where should I start? $$2x^5 + 3x^4 -10x^3 -15x^2 + 8x + 12$$

Answer 1

To approach this sort of problem, I usually proceed in one of two ways:

• The brutal approach: pose $P(x) = \prod_i (x - \alpha_i)$ and find all $\alpha_i$ at once. Note: you have to adapt this approach when the highest-order coefficient is not one.
• The fine approach: I try to check if any integer, most notably 1, -1 or 2, -2 are roots of the polynomial. If they are, then I would factorize accordingly.

In your case, 1 is a root, and so is -1. So you can start looking for a new degree-3 polynomial $Q(x)$ that satisfies $P(x) = (x-1)(x+1)Q(x)$

Then, try to the factor the rest.

Answer 2

Note, for $f(x)=2x^5+3x^4−10x^3−15x^2+8x+12$, $$f(1)=f(-1)=f(2)=f(-2)=f(-\frac{3}{2})=0$$

Using synthetic divisions with $1$ and $-1$, you get that $$f(x)=(x+1)(x-1)(2x^3+3x^2-8x-12)$$

You can use the other roots to further factor the degree-3 polynomial.