Question

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

Polynomial factorization
ANSWERED
asked 2021-05-08
Write the linear factorization and list all the zeros of \(p(x)=x^{4}+x^{3}-8x^{2}-2x+12\).

Answers (1)

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):
\(2,-3,\sqrt{2},-\sqrt{2}\).
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