Question

Let P(x)=x^{3}-3x^{2}+x-3 Find all the zeros of P. Find the complete factorization of P.

Polynomial factorization
ANSWERED
asked 2021-05-28
Let \(P(x)=x^{3}-3x^{2}+x-3\)
Find all the zeros of P.
Find the complete factorization of P.

Expert Answers (1)

2021-05-29
Step 1
Given, \(P(x)=x^{3}-3x^{2}+x-3\)
For zeroes substituting,P(x)=0
\(\Rightarrow x^{3}-3x^{2}+x-3=0\)
On simplifying further we get:
\(\Rightarrow x^{3}-3x^{2}+x-3=0\)
\(\Rightarrow x^{2}(x-3)+(x-3)=0\)
\(\Rightarrow (x-3)(x^{2}+1)=0\)
\(\Rightarrow (x-3)=0, or, (x^{2}+1)=0\)
\(\Rightarrow x=3\), or, \(x = \pm \sqrt{-1}\)
\(\Rightarrow x=3\), or, \(x=\pm i\) (using, \(\sqrt{-1}=i\))
Hence, zeroes are:x=(\(3,\pm i\))
Step 2
Required factorizes form is:
\(P(x)=x^{3}-3x^{2}+x-3=(x^{2}+1)(x-3)\)
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