Find a polynomial f(x) of degree 4 with real coefficients and the following zeros. 2 (multiplicity 2), -i.

Question

Please I need help it will be very much appreciated.

Answer 1

First, use the factored form of a polynomial. With degree 4, there will be four factors:

f(x) = a·(x-p)(x-q)(x-r)(x-s)

where a is a constant and p, q, r, s are the zeros. You are given three of the four roots. A multiplicity of 2 means that root appears twice, so the three roots are 2, 2, and -i. Now with real coefficients you can apply the Conjugate Root Theorem which tells us that if -i is a root (zero), so is +i. Now you have all 4 roots: 2, 2, -i, and i.

f(x) = a·(x-2)(x-2)(x+i)(x-i)

Since the problem asks you to find any polynomial, you are free to pick whatever value of a you want except zero. Choose a = 1 since that's the simplest:

f(x) = (x-2)(x-2)(x+i)(x-i)

Multiply it out if you want it in standard form.