Explained: "Given P(x)=3x5−5x4+37x3−83x2−176x−48, and that 4i is a zero,..."

DofotheroU

Answered question

2021-08-07

Given

P (x) = 3 x^{5} - 5 x^{4} + 37 x^{3} - 83 x^{2} - 176 x - 48

, 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) = ?

berggansS

Skilled2021-08-08Added 91 answers

Step 1
The given polynomial is:

P (x) = 3 x^{5} - 5 x^{4} + 37 x^{3} - 83 x^{2} - 176 x - 48

Step 2
Now

x = 4 i

is a zero of the given polynomial, therefore

x = - 4 i

is also zero of the given polynomial.
Therefore:

(x - 4 i) (x + 4 i)

divides the given polynomial. That is one of the factors of given polynomial is:

x^{2} + 16

\frac{P (x)}{x^{2} + 16} = \frac{3 x^{5} - 5 x^{4} + 37 x^{3} - 83 x^{2} - 176 x - 48}{x^{2} + 16}

\frac{P (x)}{x^{2} + 16} = 3 x^{3} - 5 x^{2} - 11 x - 3

P (x) = (x^{2} + 16) (3 x^{3} - 5 x^{2} - 11 x - 3)

P (x) = (x^{2} + 16) (x + 1) (x - 3) (3 x + 1)

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