Find a polynomial function ƒ(x) of least degree with real coefficients having zeros as given. √3, -√3, 2, 3

Verified step by step guidance

1. Identify the zeros of the polynomial function: \( \sqrt{3}, -\sqrt{3}, 2, 3 \). Since the polynomial has real coefficients, these zeros can be used directly to form factors.
2. Write the factors corresponding to each zero. For a zero \( r \), the factor is \( (x - r) \). So the factors are \( (x - \sqrt{3}), (x + \sqrt{3}), (x - 2), (x - 3) \).
3. Group the conjugate pair \( (x - \sqrt{3}) \) and \( (x + \sqrt{3}) \) to form a quadratic factor by multiplying them: \( (x - \sqrt{3})(x + \sqrt{3}) = x^2 - (\sqrt{3})^2 = x^2 - 3 \).
4. Write the polynomial function as the product of the quadratic factor and the remaining linear factors: \( f(x) = (x^2 - 3)(x - 2)(x - 3) \).
5. Expand the factors step-by-step if needed: first multiply \( (x - 2)(x - 3) \), then multiply the result by \( (x^2 - 3) \) to get the polynomial in standard form.

Key Concepts

Polynomial Zeros and Factors

Each zero of a polynomial corresponds to a factor of the form (x - zero). For example, if √3 is a zero, then (x - √3) is a factor. To find the polynomial, multiply all factors corresponding to the given zeros.

Real Coefficients and Conjugate Pairs

When a polynomial has real coefficients, irrational zeros like √3 must appear in conjugate pairs (√3 and -√3) to ensure the product results in real coefficients. This guarantees the polynomial remains with real numbers only.

Least Degree Polynomial

The polynomial of least degree that has the given zeros is formed by using each zero exactly once as a factor. This ensures the polynomial is the simplest (lowest degree) that satisfies the zero conditions.