Question

Suppose f(x) is a 4th degree polynomial with real coefficients and roots x = 2, x = -1, and -3 - 2i. Find the remaining root of f(x).

Answer 1

The remaining root of the 4th degree polynomial with the given roots is -3 + 2i, since complex roots must come in conjugate pairs when the coefficients are real.

Step-by-step explanation:

If f(x) is a 4th degree polynomial with real coefficients and roots x = 2, x = -1, and -3 - 2i, to find the remaining root of f(x), we must recognize that non-real roots of polynomials with real coefficients come in conjugate pairs. Thus, if -3 - 2i is a root, then its conjugate, -3 + 2i, must also be a root. The remaining root of the polynomial is -3 + 2i. This is because the polynomial's coefficients are real, and complex roots always come in pairs, known as complex conjugates. To write the polynomial in factored form, we would have f(x) = (x - 2)(x + 1)(x + 3 + 2i)(x + 3 - 2i).