SOLUTION: Form a polynomial f left parenthesis x right parenthesis with real coefficients having the given degree and zeros. Degree​ 4; ​ zeros: 3+2i ; -5 multiplicity 2 Enter the pol

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Form a polynomial f left parenthesis x right parenthesis with real coefficients having the given degree and zeros. Degree​ 4; ​ zeros: 3+2i ; -5 multiplicity 2 Enter the pol      Log On


   

Question 1157331: Form a polynomial f left parenthesis x right parenthesis with real coefficients having the given degree and zeros.
Degree​ 4; ​ zeros: 3+2i ; -5 multiplicity 2
Enter the polynomial.
f(x)=a(___)
Answer by greenestamps(13334) About Me  (Show Source):
You can put this solution on YOUR website!


The polynomial is to have real coefficients, so the complex zeros occur in conjugate pairs. So the roots are 3+2i, 3-2i, -5, and -5.

Use Vieta's theorem to find the quadratic polynomial with roots 3+2i and 3-2i:
%283%2B2i%29%2B%283-2i%29+=+6
%283%2B2i%29%283-2i%29+=+9-%28-4%29+=+13

The quadratic polynomial is

x%5E2-6x%2B13

The quadratic polynomial with roots -5 and -5 is

x%5E2%2B10x%2B25

The polynomial we want is

f%28x%29+=+%28x%5E2-6x%2B13%29%28x%5E2%2B10x%2B25%29+=+x%5E4%2B4x%5E3-22x%5E2-20x%2B325