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

Verified step by step guidance

1. Identify the given zeros of the polynomial: \$2 - i\(, \)3\(, and \)-1\(. Since the polynomial must have only real coefficients, the complex zeros must come in conjugate pairs. Therefore, include the conjugate of \)2 - i\(, which is \)2 + i$, as a zero as well.
2. Write the factors corresponding to each zero. For zeros \$3\( and \)-1\(, the factors are \)(x - 3)\( and \)(x + 1)\( respectively. For the complex zeros \)2 - i\( and \)2 + i\(, the factors are \)(x - (2 - i))\( and \)(x - (2 + i))$.
3. Multiply the complex conjugate factors to get a quadratic with real coefficients: \[ (x - (2 - i))(x - (2 + i)) = ((x - 2) + i)((x - 2) - i) = (x - 2)^2 - (i)^2 \]
4. Simplify the expression from the previous step using \(i^2 = -1\): \[ (x - 2)^2 - (-1) = (x - 2)^2 + 1 \]
5. Write the polynomial function \(f(x)\) as the product of the quadratic from step 4 and the linear factors from step 2: \[ f(x) = ((x - 2)^2 + 1)(x - 3)(x + 1) \]

Key Concepts

Complex Conjugate Root Theorem

For polynomials with real coefficients, any non-real complex roots must occur in conjugate pairs. Since 2 - i is a root, its conjugate 2 + i must also be a root to ensure the polynomial has real coefficients.

Constructing Polynomials from Roots

A polynomial can be formed by creating factors from each root, such as (x - root). Multiplying these factors together yields a polynomial with the given roots. The degree of the polynomial equals the number of roots, counting multiplicities.

Multiplicity of Roots

Multiplicity refers to how many times a root appears. Here, all roots have multiplicity 1 unless stated otherwise, meaning each root contributes one factor to the polynomial. This affects the degree and shape of the polynomial.