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.

aortiH 2021-08-07 Answered
Given P(x)=3x58x4+35x398x2208x+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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Expert Answer

hosentak
Answered 2021-08-08 Author has 100 answers
Step 1
Given: P(x)=3x58x4+35x398x2208x+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, (x4i),(x+4i) are factors of P(x)
So, (x4i)(x+4i) is a factor of P(x)
x2+16 is a factor of P(x)
Dividing P(x) by x2+16
So, 3x58x4+35x398x2208x+480x2+16=3x38x213x+30
P(x)=(3x38x213x+30)(x2+16)
Step 2
Consider F(x)=3x38x213x+30
We observe that F(2)=3(2)38(2)213(2)+30
F(2)=2432+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+4i4i+(2)+(3)=83
α=831
α=53
Hence, the zeros are 2,3,4i,4i,53
Hence, P(x)=3(x+2)(x3)(x4i)(x+4i)(x53)
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