find all zeros of $f(x)=x^{3}-x^{2}+x - 1$. enter the zeros separated by commas.
find all zeros of $f(x)=x^{3}-x^{2}+x - 1$. enter the zeros separated by commas.
Answer
Explanation:
Step1: Factor the polynomial by grouping
Group the terms of $f(x)=x^{3}-x^{2}+x - 1$ as $(x^{3}-x^{2})+(x - 1)$. Then factor out the common factors from each group: $x^{2}(x - 1)+1(x - 1)=(x - 1)(x^{2}+1)$.
Step2: Set each factor equal to zero
Set $x - 1=0$ and $x^{2}+1=0$. For $x - 1=0$, we get $x = 1$. For $x^{2}+1=0$, we can rewrite it as $x^{2}=-1$. Then $x=\pm i$.
Answer:
$1,i,-i$