Given $P\left(x\right)=3{x}^{5}-8{x}^{4}+35{x}^{3}-98{x}^{2}-208x+480$ , and that 4i is a zero, write P in factored form (as a product of linear factors). Be sure to write the full equation, including $P\left(x\right)=$

aortiH
2021-08-07
Answered

Given $P\left(x\right)=3{x}^{5}-8{x}^{4}+35{x}^{3}-98{x}^{2}-208x+480$ , and that 4i is a zero, write P in factored form (as a product of linear factors). Be sure to write the full equation, including $P\left(x\right)=$

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hosentak

Answered 2021-08-08
Author has **100** answers

Step 1

Given:$P\left(x\right)=3{x}^{5}-8{x}^{4}+35{x}^{3}-98{x}^{2}-208x+480$

It is given that 4i is a zero of P(x)

We know that for polynomials with real coefficients, the complex zeros always exists in conjugate pairs

Hence, -4i is also a zero of P(x)

Hence,$(x-4i),(x+4i)$ are factors of P(x)

So,$(x-4i)(x+4i)$ is a factor of P(x)

$\Rightarrow {x}^{2}+16$ is a factor of P(x)

Dividing P(x) by${x}^{2}+16$

So,$\frac{3{x}^{5}-8{x}^{4}+35{x}^{3}-98{x}^{2}-208x+480}{{x}^{2}+16}=3{x}^{3}-8{x}^{2}-13x+30$

$P\left(x\right)=(3{x}^{3}-8{x}^{2}-13x+30)({x}^{2}+16)$

Step 2

Consider$F\left(x\right)=3{x}^{3}-8{x}^{2}-13x+30$

We observe that$F(-2)=3{(-2)}^{3}-8{(-2)}^{2}-13(-2)+30$

$F(-2)=-24-32+26+30$

$F(-2)=0$

Similarly,$F\left(3\right)=0$

So, -2,3 are zeros of F(x) and hence of P(x) too

Let$\alpha$ be the fifth zero of P(x)

We know that the sum of zeros of$P\left(x\right)=\frac{8}{3}$

$\alpha +4i-4i+(-2)+\left(3\right)=\frac{8}{3}$

$\alpha =\frac{8}{3}-1$

$\alpha =\frac{5}{3}$

Hence, the zeros are$-2,3,4i,-4i,\frac{5}{3}$

Hence,$P\left(x\right)=3(x+2)(x-3)(x-4i)(x+4i)(x-\frac{5}{3})$

Given:

It is given that 4i is a zero of P(x)

We know that for polynomials with real coefficients, the complex zeros always exists in conjugate pairs

Hence, -4i is also a zero of P(x)

Hence,

So,

Dividing P(x) by

So,

Step 2

Consider

We observe that

Similarly,

So, -2,3 are zeros of F(x) and hence of P(x) too

Let

We know that the sum of zeros of

Hence, the zeros are

Hence,

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