SOLUTION: Find a polynomial function of degree 3 with zeros 1/2, 1-i; where P(4) = 140

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find a polynomial function of degree 3 with zeros 1/2, 1-i; where P(4) = 140      Log On


   



Question 682329: Find a polynomial function of degree 3 with zeros 1/2, 1-i; where P(4) = 140
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Find a polynomial function of degree 3 with zeros 1/2, 1-i; where P(4) = 140
Since the polynomial is to have real coefficients, since 1-i is a zero,
its conjugate 1+i is also a zero

So the polynomial is

%22P%28x%29%22%22%22=%22%22A%2A%28x-1%2F2%29%28x-%281-i%29%29%28x-%281%2Bi%29%29
%22P%28x%29%22%22%22=%22%22A%2A%28x-1%2F2%29%28x-1%2Bi%29%28x-1-i%29
%22P%28x%29%22%22%22=%22%22A%2A%28x-1%2F2%29%28%28x-1%29%2Bi%29%28%28x-1%29-i%29
%22P%28x%29%22%22%22=%22%22A%2A%28x-1%2F2%29%28%28x-1%29%5E2-i%5E2%29
%22P%28x%29%22%22%22=%22%22A%2A%28x-1%2F2%29%28%28x-1%29%5E2-%28-1%29%29
%22P%28x%29%22%22%22=%22%22A%2A%28x-1%2F2%29%28%28x-1%29%5E2%2B1%29
%22P%28x%29%22%22%22=%22%22A%2A%282x%2F2-1%2F2%29%28%28x%5E2-2x%2B1%29%2B1%29
%22P%28x%29%22%22%22=%22%22A%2A%28%282x-1%29%2F2%29%28x%5E2-2x%2B1%2B1%29
%22P%28x%29%22%22%22=%22%22A%2A%28%282x-1%29%2F2%29%28x%5E2-2x%2B2%29
2%2A%22P%28x%29%22%22%22=%22%22A%282x-1%29%28x%5E2-2x%2B2%29
2%2A%22P%28x%29%22%22%22=%22%22A%2A%282x%5E3-4x%5E2%2B4x-x%5E2%2B2x-2%29
2%2A%22P%28x%29%22%22%22=%22%22A%2A%282x%5E3-5x%5E2%2B6x-2%29

Now we substitute %22P%284%29%22%22%22=%22%22140, 
which implies that we also substitutex%22%22=%22%224.

2%2A%22P%284%29%22%22%22=%22%22A%282x%2A4%5E3-5%2A4%5E2%2B6%2A4-2%29
2%2A140%22%22=%22%22A%2A%282%2A64-5%2A16%2B6%2A4-2%29
280%22%22=%22%22A%2A%28128-80%2B24-2%29
280%22%22=%22%22A%2A%2870%29
280%2F70%22%22=%22%22A
4%22%22=%22%22A

So

2%2A%22P%28x%29%22%22%22=%22%224%2A%282x%5E3-5x%5E2%2B6x-2%29
%22P%28x%29%22%22%22=%22%222%2A%282x%5E3-5x%5E2%2B6x-2%29
%22P%28x%29%22%22%22=%22%224x%5E3-10x%5E2%2B12x-4

Edwin