Question

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

7, -11, and 2 + 8i

Answer 1

A key point:

"real coefficients" — this means any complex zeros must come in conjugate pairs, because that is the only way to get real coefficients with complex zeros.

Answer 2

If you have a polynomial in \(x\) with zeros \((r_1, r_2, ... r_n\) you can construct it by expanding the product of factors \[(x-r_1)(x-r_2)...(x-r_n)\]

Your complex zeros will give you pairs of zeros: \(a\pm bi\) where \(i = \sqrt{-1}\) and their factors will be \((x-(a+bi))(x-(a-bi))\)

When expanding, it's often easier to multiply the terms with complex zeros first, so that you simplify away the mess before doing more multiplication.