SOLUTION: find the equation for p p is a degree 5 polynomial that passes through the origin, has zeros i and 4−i, and p(5)=520

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Question 1161068: find the equation for p p is a degree 5 polynomial that passes through the origin, has zeros i and 4−i, and p(5)=520
Found 3 solutions by greenestamps, MathLover1, ikleyn:
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


The graph passing through the origin means one root is 0; the polynomial has a factor of x.

A zero of i means a zero of -i. The quadratic factor producing those two roots is x%5E2%2B1

A zero of 4-i means a zero of 4+1. The quadratic factor producing those two roots is x%5E2-8x%2B17

The polynomial is

a%28x%29%28x%5E2%2B1%29%28x%5E2-8x%2B17%29

The constant a is determined by the requirement that p(5) = 520.

a%285%29%2826%29%282%29+=+520
260a+=+520
a+=+2

The polynomial function is

2%28x%29%28x%5E2%2B1%29%28x%5E2-8x%2B17%29

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

given:
p%28x%29 is a degree+5

has zeros :
x%5B1%5D=i+-> complex zeros always come in pair, so you also have
x%5B2%5D=-i+
x%5B3%5D=4-i+, and also
x%5B4%5D=4%2Bi+
passes through the origin: point (0,0)-> 5th root is x%5B5%5D=0
p%285%29=520

use zero product rule:
p%28x%29=a%28x-x%5B1%5D%29%28x-x%5B2%5D%29%28x-x%5B3%5D%29%28x-x%5B4%5D%29%28x-x%5B5%5D%29....substitute given zeros

p%28x%29=a%28x-i%29%28x-%28-i%29%29%28x-%284-i+%29%29%28x-%284%2Bi+%29%29%28x-0%29
p%28x%29=a%28x%5E5+-+8x%5E4+%2B+18x%5E3+-+8x%5E2+%2B+17x%29......since we are givenp%285%29=520, use it to find a
520=a%285%5E5+-+8%2A5%5E4+%2B+18%2A5%5E3+-+8%2A5%5E2+%2B+17%2A5%29
520=a%28260%29
a=520%2F260
a=2
and your equation is
p%28x%29=2%28x%5E5+-+8x%5E4+%2B+18x%5E3+-+8x%5E2+%2B+17x%29
p%28x%29=2x%5E5+-+16x%5E4+%2B+36x%5E3+-+16x%5E2+%2B+34x+


Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.

As the problem is presented in the post, it is a FAULT,

since the post does not state that the polynomial is with real coefficients.


At the university, where I studied Math, for such formulation the student could deserve only lowest possible score.

It only means that the person who created it, does not know Math and does not know this subject, in particular.