Question

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form.

3, -13, and 5 + 4i

A. f(x) = x4 - 8x3 - 12x2 + 400x - 1599

B. f(x) = x4 - 200x2 + 800x - 1599

C. f(x) = x4 - 98x2 + 800x - 1599

D. f(x) = x4 - 8x3 + 12x2 - 400x + 1599

Answer 1

We have roots of x = 3, -13, and 5+4i. We're going to need a conjugate to 5+4i also, because the complex roots come in pairs, so that will be 5-4i. Now the roots are simply the places where the factors of the polynomial = 0, so we can write our factored polynomial as \[f(x) = (x-3)(x+13)(x-5-4i)(x-5+4i) = 0\] Now it's just some pencil-pushing to multiply that expression out to get f(x) = x^4 + more terms. I would suggest multiplying the (x-5-4i) and (x-5+4i) terms first, as doing so will eliminate the terms containing i (remember that i^2 = -1).