# Find all zeros of the polynomial. P(x) = x^{5} - 2x^{4}

Find all zeros of the polynomial.
$P\left(x\right)={x}^{5}-2{x}^{4}+2{x}^{3}-4{x}^{2}+x-2$
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aquariump9
Step 1
Given,
$P\left(x\right)={x}^{5}-2{x}^{4}+2{x}^{3}-4{x}^{2}+x-2$
now find the factors and zeros of function
p(x)=0
${x}^{4}\left(x-2\right)+2{x}^{2}\left(x-2\right)+1\left(x-2\right)$
$\left(x-2\right)\left({x}^{4}+2{x}^{2}+1\right)=0$
now find factor of ${x}^{4}+2{x}^{2}+1=0$
$\left(x-2\right){\left({x}^{2}+1\right)}^{2}=0$
x-2=0
x=2
Step 2
${x}^{2}+1=0$
$x=±\sqrt{-1}=±i$
so polynomial has only one real zeros at x=2
and two complex zeros.