Question

# Write the linear factorization and list all the zeros of p(x)=x^{4}+x^{3}-8x^{2}-2x+12.

Polynomial factorization
Write the linear factorization and list all the zeros of $$p(x)=x^{4}+x^{3}-8x^{2}-2x+12$$.

2021-05-09
Step 1
$$p(x)=x^{4}+x^{3}-8x^{2}-2x+12$$
$$\Rightarrow p(x)=x^{3}(x-2)+3x^{2}(x-2)-2x(x-2)-6(x-2)$$
$$\Rightarrow p(x)=(x-2)(x^{3}+3x^{2}-2x-6)$$
$$\Rightarrow p(x)=(x-2)[x^{2}(x+3)-2(x+3)]$$
$$\Rightarrow p(x)=(x-2)(x+3)(x^{2}-2)$$
$$\Rightarrow p(x)=(x-2)(x+3)(x+\sqrt{2})(x-\sqrt{2})$$
Step 2
linear factorization of p(x).
$$p(x)=(x-2)(x+3)(x+\sqrt{2})(x-\sqrt{2})$$
Zeros of p(x):
$$2,-3,\sqrt{2},-\sqrt{2}$$.