Given P(x)=3x5−8x4+35x3−98x2−208x+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(x)=

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Step 1  Given: P(x)=3x5−8x4+35x3−98x2−208x+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)  ⇒x2+16 is a factor of P(x)  Dividing P(x) by x2+16  So, 3x5−8x4+35x3−98x2−208x+480x2+16=3x3−8x2−13x+30  P(x)=(3x3−8x2−13x+30)(x2+16)  Step 2  Consider F(x)=3x3−8x2−13x+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(3)=0  So, -2,3 are zeros of F(x) and hence of P(x) too  Let α be the fifth zero of P(x)  We know that the sum of zeros of P(x)=83  α+4i−4i+(−2)+(3)=83  α=83−1  α=53  Hence, the zeros are −2,3,4i,−4i,53  Hence, P(x)=3(x+2)(x−3)(x−4i)(x+4i)(x−53)

Given P(x)=3x^{5}-8x^{4}+35x^{3}-98x^{2}-208x+480, and that 4i is a zero, write P in factored form be sure to write the full equation.

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