Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. 5+i and 5-i

Verified step by step guidance

1. Identify the given zeros of the polynomial: 5+i and 5-i. Since the polynomial must have real coefficients, complex zeros occur in conjugate pairs, which is already the case here.
2. Write the factors corresponding to each zero. For zero 5+i, the factor is \(\left(x - (5+i)\right)\), and for zero 5-i, the factor is \(\left(x - (5-i)\right)\).
3. Multiply the two factors to form a quadratic polynomial: \(\left(x - (5+i)\right)\left(x - (5-i)\right)\).
4. Use the difference of squares formula to simplify the product: \(\left(x - 5 - i\right)\left(x - 5 + i\right) = \left(x - 5\right)^2 - (i)^2\).
5. Simplify further by substituting \(i^2 = -1\), resulting in \(\left(x - 5\right)^2 + 1\). This is the polynomial function of least degree with real coefficients having the given zeros.

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 the polynomial remains with real coefficients when expanded.

Forming Polynomial from Roots

A polynomial can be constructed by creating factors from its roots. For roots r1 and r2, the polynomial includes factors (x - r1) and (x - r2). Multiplying these factors yields a polynomial with those roots.

Least Degree Polynomial

The polynomial of least degree containing given roots has exactly those roots and no others, with multiplicity one unless specified. This means the polynomial's degree equals the number of distinct roots.