Question

# Find the complete factorization of P(x)= x^{4} - 2x^{3} + 5x^{2} - 8x + 4.

Polynomial factorization
Find the complete factorization of $$P(x)= x^{4} - 2x^{3} + 5x^{2} - 8x + 4$$.

2021-06-04

Step 1
Given:
$$x^{4} - 2x^{3} + 5x^{2} - 8x + 4$$
Step 2
$$x^{4} - 2x^{3} + 5x^{2} - 8x + 4$$
Let $$p(x) = x^{4}-2x^{3}+5x^{2}-8x+4$$
let $$x = 1 \Rightarrow x-1=0$$
$$p(1)=(1)^{4}-2(1)^{3}+5(1)^{2}-8(1)+4$$
$$p(1)=1-2+5-8+4$$
$$p(1)=0$$
$$\Rightarrow x-1$$ is the factor of p(x)
$$x^{4}-2x^{3}+5x^{2}-8x+4=(x-1)(x^{3}-x^{2}+4x-4)$$
$$x^{4}-2x^{3}+5x^{2}-8x+4=(x-1)(x^{2}(x-1)+4(x-1))$$
$$x^{4}-2x^{3}+5x^{2}-8x+4=(x-1)(x^{2}+4)(x-1)$$
$$x^{4}-2x^{3}+5x^{2}-8x+4=(x-1)^{2}(x^{2}+4)$$