Question

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

Polynomial factorization
ANSWERED
asked 2021-06-03
Find the complete factorization of \(P(x)= x^{4} - 2x^{3} + 5x^{2} - 8x + 4\).

Answers (1)

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)
\(x^{4}-2x^{3}+5x^{2}-8x+4=(x-1)(x^{3}-x^{2}+4x-4)\)
\(x^{4}-2x^{3}+5x^{2}-8x+4=(x-1)(x^{2}(x-1)+4(x-1))\)
\(x^{4}-2x^{3}+5x^{2}-8x+4=(x-1)(x^{2}+4)(x-1)\)
\(x^{4}-2x^{3}+5x^{2}-8x+4=(x-1)^{2}(x^{2}+4)\)

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