SOLUTION: The degree three polynomial f(x) with real coefficients and leading coefficient 1, has -3 and +4i among its roots. Express f(x) as a product of linear and quadratic polynomials wi

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: The degree three polynomial f(x) with real coefficients and leading coefficient 1, has -3 and +4i among its roots. Express f(x) as a product of linear and quadratic polynomials wi      Log On


   

Question 118101: The degree three polynomial f(x) with real coefficients and leading coefficient 1, has -3 and +4i among its roots. Express f(x) as a product of linear and quadratic polynomials with real coefficients.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Remember, if a polynomial has roots of a and b, then it's factorization will be %28x-a%29%28x-b%29


So if a polynomial has roots of -3, 4i, and -4i (remember complex roots always come in pairs) , then it's factorization will be

%28x-%28-3%29%29%28x-4i%29%28x-%28-4i%29%29


%28x%2B3%29%28x-4i%29%28x%2B4i%29 Rewrite x-%28-3%29 as x%2B3 and x-%28-4i%29 as x%2B4i


%28x%2B3%29%28x%5E2%2B4xi-4xi-4i%5E2%29 Foil %28x-4i%29%28x%2B4i%29 to get x%5E2%2B4xi-4xi-4i%5E2


%28x%2B3%29%28x%5E2-16i%5E2%29 Combine like terms


%28x%2B3%29%28x%5E2-16%28-1%29%29 Replace i%5E2 with -1


%28x%2B3%29%28x%5E2%2B16%29 Multiply



x%5E3%2B3x%5E2%2B16x%2B48 Now foil %28x%2B3%29%28x%5E2%2B16%29 to get x%5E3%2B3x%5E2%2B16x%2B48




So the polynomial with roots of -3 and +4i is x%5E3%2B3x%5E2%2B16x%2B48