Ernstfalld

2020-11-30

Complete Factorization Factor the polynomial completely, and find all its zeros.State the multiplicity of each zero.
$P\left(x\right)={x}^{5}+7{x}^{3}$

broliY

Concept used:
The multiplicity of zero of the polynomial having factor $\left(x—c\right)$ that appears k times in the factorization of the polynomial is k.
Calculation:
The given polynomial is $P\left(x\right)={x}^{5}+7{x}^{3}$.
Factor the above polynomial to obtain the zeros.
$P\left(x\right)={x}^{5}+7{x}^{3}$
$={x}^{3}\left({x}^{2}+7\right)$
$={x}^{3}\left({x}^{2}-\left(\sqrt{7i}{\right)}^{2}\right)$
$={x}^{3}\left(x-\sqrt{7i}\right)\left(x+\sqrt{7i}\right)$
Substitute 0 for P (x) in the polynomial $P\left(x\right)={x}^{5}+7{x}^{3}$ to obtain the zeros of the polynomial
. ${x}^{3}\left(x-\sqrt{7i}\right)\left(x+\sqrt{7i}\right)$
Further solve for the value of x as,
${x}^{3}=0,\left(x-\sqrt{7i}\right)=0$ and $\left(x+\sqrt{7i}\right)=0$
$x=0,x=\sqrt{7i}$ and $x=-\sqrt{7i}$
The zeros of the polynomial $P\left(x\right)={x}^{5}+7{x}^{3}$ appears three times and one time in the polynomial therefore, the multiplicity of the zero 0 is 3, $\sqrt{7i}$ and $\sqrt{7i}$ is 1.
Conclusion:
Thus, the factorization of the polynomial $P\left(x\right)={x}^{5}+7{x}^{3}$ is $P\left(x\right)={x}^{3}\left(x-\sqrt{7i}\right)\left(x+\sqrt{7i}\right)$, zeros of the polynomial are 0 and + $\sqrt{7i}$ and the multiplicity of the zero 0 is 3, is 1.

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