If ƒ(x) is a polynomial function with real coefficients, and if 7+2i is a zero of the function, then what other complex number must also be a zero?

Verified step by step guidance

1. Recall that if a polynomial function has real coefficients, then any non-real complex zeros must occur in conjugate pairs.
2. Identify the given zero: it is the complex number \(7 + 2i\), where \(i\) is the imaginary unit.
3. Find the complex conjugate of the given zero by changing the sign of the imaginary part: the conjugate of \(7 + 2i\) is \(7 - 2i\).
4. Conclude that since \(7 + 2i\) is a zero of the polynomial with real coefficients, its conjugate \(7 - 2i\) must also be a zero.
5. This property ensures that the polynomial's coefficients remain real when expanded from its factors.

Key Concepts

Complex Conjugate Root Theorem

This theorem states that if a polynomial has real coefficients and a complex number a + bi is a root, then its conjugate a - bi must also be a root. This ensures that non-real roots come in pairs, maintaining real coefficients in the polynomial.

Polynomial Functions with Real Coefficients

A polynomial function with real coefficients means all the coefficients of the terms are real numbers. This property influences the nature of the roots, particularly ensuring that complex roots appear in conjugate pairs to keep the polynomial's coefficients real.

Complex Numbers and Their Conjugates

A complex number is expressed as a + bi, where a and b are real numbers and i is the imaginary unit. Its conjugate is a - bi, which reflects the number across the real axis in the complex plane. Conjugates have the same real part but opposite imaginary parts.