Ernstfalld

2020-11-30

Complete Factorization Factor the polynomial completely, and find all its zeros.State the multiplicity of each zero.

$P(x)={x}^{5}+7{x}^{3}$

broliY

Skilled2020-12-01Added 97 answers

Concept used:

The multiplicity of zero of the polynomial having factor$(x\u2014c)$ that appears k times in the factorization of the polynomial is k.

Calculation:

The given polynomial is$P(x)={x}^{5}+7{x}^{3}$ .

Factor the above polynomial to obtain the zeros.

$P(x)={x}^{5}+7{x}^{3}$

$={x}^{3}({x}^{2}+7)$

$={x}^{3}({x}^{2}-(\sqrt{7i}{)}^{2})$

$={x}^{3}(x-\sqrt{7i})(x+\sqrt{7i})$

Substitute 0 for P (x) in the polynomial$P(x)={x}^{5}+7{x}^{3}$ to obtain the zeros of the polynomial

.${x}^{3}(x-\sqrt{7i})(x+\sqrt{7i})$

Further solve for the value of x as,

${x}^{3}=0,(x-\sqrt{7i})=0$ and $(x+\sqrt{7i})=0$

$x=0,x=\sqrt{7i}$ and $x=-\sqrt{7i}$

The zeros of the polynomial$P(x)={x}^{5}+7{x}^{3}$ 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$P(x)={x}^{5}+7{x}^{3}$ is $P(x)={x}^{3}(x-\sqrt{7i})(x+\sqrt{7i})$ , zeros of the polynomial are 0 and + $\sqrt{7i}$ and the multiplicity of the zero 0 is 3, $\sqrt{7i}\text{}and\text{}-\sqrt{7i}$ is 1.

The multiplicity of zero of the polynomial having factor

Calculation:

The given polynomial is

Factor the above polynomial to obtain the zeros.

Substitute 0 for P (x) in the polynomial

.

Further solve for the value of x as,

The zeros of the polynomial

Conclusion:

Thus, the factorization of the polynomial

$\frac{20b}{{\left(4{b}^{3}\right)}^{3}}$

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