Which statement about non-real roots for polynomials with real coefficients is true?

Which statement about non-real roots for polynomials with real coefficients is true?

• A. They occur singly
• B. They occur in conjugate pairs
• C. They must be real
• D. They occur in triples

Answer: (B)

When a polynomial has real coefficients, complex roots come in conjugate pairs. If a + bi (with b ≠ 0) is a root, then the conjugate a − bi is also a root. This happens because the polynomial with real coefficients satisfies P(z) = sum real_coef_k z^k, so taking complex conjugates gives P(a − bi) = overline(P(a + bi)) = 0 whenever P(a + bi) = 0. This pairing explains why non-real roots never occur alone or in triples: they always appear together as a conjugate pair. The quadratic factor x^2 − 2ax + (a^2 + b^2) corresponds to such a pair and has real coefficients, reinforcing why these roots must come in pairs.

When a polynomial has real coefficients, complex roots come in conjugate pairs. If a + bi (with b ≠ 0) is a root, then the conjugate a − bi is also a root. This happens because the polynomial with real coefficients satisfies P(z) = sum real_coef_k z^k, so taking complex conjugates gives P(a − bi) = overline(P(a + bi)) = 0 whenever P(a + bi) = 0. This pairing explains why non-real roots never occur alone or in triples: they always appear together as a conjugate pair. The quadratic factor x^2 − 2ax + (a^2 + b^2) corresponds to such a pair and has real coefficients, reinforcing why these roots must come in pairs.