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

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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.

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 \((x - (2 + i))\).

Multiply the complex conjugate factors to get a quadratic with real coefficients: \[ (x - (2 - i))(x - (2 + i)) = ((x - 2) + i)((x - 2) - i) = (x - 2)^2 - (i)^2 \]

Simplify the expression from the previous step using \(i^2 = -1\): \[ (x - 2)^2 - (-1) = (x - 2)^2 + 1 \]

Write the polynomial function \(f(x)\) as the product of the quadratic from step 4 and the linear factors from step 2: \[ f(x) = ((x - 2)^2 + 1)(x - 3)(x + 1) \]

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.