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

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


   

Question 114557: The degree three polynomial f(x) with real coefficients and leading coefficient 1, has 4 and 3 + i among its roots. Express f(x) as a product of linear and quadratic polynomials with real coefficients.
Answer by solver91311(24713) About Me  (Show Source):
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If one of the roots is 3%2Bi, then 3-i must also be a root because complex roots come in conjugate pairs a%2B-bi.

Now that we know all three roots, namely 4, 3%2Bi, and 3-i, we can create a linear binomial factor and a quadratic factor that will represent the desired degree three polynomial.

Factor 1: %28x-4%29

Factor 2: %28x-%283%2Bi%29%29%28x-%283-i%29%29
x%5E2-%283-i%29x-%283%2Bi%29x%2B9%2B1
x%5E2-3x%2Bix-3x-ix%2B10
x%5E2-6x%2B10

Complete the expression for f(x)
f%28x%29=%28x-4%29%28x%5E2-6x%2B10%29