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, 3, and -1

Verified step by step guidance

1. Identify the given zeros: \(2-i\), \(3\), and \(-1\).
2. Since the polynomial has real coefficients, the complex zero \(2-i\) must have its conjugate \(2+i\) as a zero as well.
3. Write the factors corresponding to each zero: \((x - (2-i))\), \((x - (2+i))\), \((x - 3)\), and \((x + 1)\).
4. Multiply the conjugate pair factors: \((x - (2-i))(x - (2+i))\) to get a quadratic factor.
5. Multiply all the factors together: \((x - 3)(x + 1)\) with the quadratic factor obtained in the previous step to form the polynomial \(f(x)\).

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 given by 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 representing the degree of the polynomial.

Zeros of a Polynomial

The zeros (or roots) of a polynomial are the values of x for which the polynomial evaluates to zero. For a polynomial of degree n, there can be up to n zeros, and each zero corresponds to a factor of the polynomial. If a zero has a multiplicity greater than one, it indicates that the factor associated with that zero is repeated in the polynomial's factorization.

Complex Conjugate Root Theorem

The Complex Conjugate Root Theorem states that if a polynomial has real coefficients and a complex number is a root, then its complex conjugate must also be a root. This is crucial when determining the polynomial's factors, as it ensures that all roots can be expressed in terms of real coefficients, which is necessary for constructing the polynomial function with the given zeros.