Skill: Factor higher-degree polynomials

When a polynomial has degree $3$ or greater, look for rational zeros first. If $r$ is a zero of $P(x)$, then $(x-r)$ is a factor.

Consider

$$P(x)=x^4-2x^3-7x^2+8x+12.$$

1. List possible rational zeros

Because the leading coefficient is $1$, the Rational Root Theorem says that any rational zero must be a factor of the constant term $12$:

$$\pm1,\ \pm2,\ \pm3,\ \pm4,\ \pm6,\ \pm12.$$

Test a simple candidate:

$$P(2)=16-16-28+16+12=0.$$

Therefore, $x=2$ is a zero, so $(x-2)$ is a factor.

2. Divide by the factor

Use synthetic division:

Thus,

$$P(x)=(x-2)(x^3-7x-6).$$

3. Factor the remaining polynomial

Test another possible rational zero in the cubic:

$$3^3-7(3)-6=27-21-6=0.$$

So $x=3$ is a zero, and $(x-3)$ is a factor. Dividing gives

$$x^3-7x-6=(x-3)(x^2+3x+2).$$

Now factor the quadratic:

$$x^2+3x+2=(x+1)(x+2).$$

Therefore, the complete factorization is

$$\boxed{x^4-2x^3-7x^2+8x+12=(x-2)(x-3)(x+1)(x+2)}.$$

Always check by multiplying the factors back together. The product expands to the original polynomial, confirming that the factorization is correct.