Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. See Examples 4–6. 2-i and 6-3i

Verified step by step guidance

1. Identify the given zeros: \(2-i\) and \(6-3i\).
2. Since the polynomial has real coefficients, the complex conjugates must also be zeros: \(2+i\) and \(6+3i\).
3. Write the factors corresponding to each zero: \((x - (2-i))\), \((x - (2+i))\), \((x - (6-3i))\), and \((x - (6+3i))\).
4. Use the difference of squares formula to simplify the product of conjugate pairs: \((x - (2-i))(x - (2+i)) = (x-2)^2 + 1\) and \((x - (6-3i))(x - (6+3i)) = (x-6)^2 + 9\).
5. Multiply the simplified factors to find the polynomial function \(f(x)\).

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable x is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n, a_(n-1), ..., a_0 are constants and n is a non-negative integer. Understanding polynomial functions is essential for constructing and analyzing their properties, such as degree and zeros.

Complex Conjugate Root Theorem

The Complex Conjugate Root Theorem states that if a polynomial has real coefficients, then any non-real complex roots must occur in conjugate pairs. For example, if 2-i is a root, then its conjugate 2+i must also be a root. This theorem is crucial for determining all roots of a polynomial when given some complex roots, ensuring that the polynomial remains with real coefficients.

Finding Polynomial from Roots

To construct a polynomial function from its roots, one can use the fact that if r is a root, then (x - r) is a factor of the polynomial. For multiple roots, the factor is raised to the power of the root's multiplicity. By multiplying the factors corresponding to all given roots, including complex conjugates, one can derive the polynomial function of least degree that satisfies the conditions of the problem.