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: \( \sqrt{3}, -\sqrt{3}, 2, \) and \( 3 \).
2. Since the polynomial has real coefficients, each zero must appear in conjugate pairs if they are complex. Here, all zeros are real, so no need for conjugate pairs.
3. Write the factors corresponding to each zero: \((x - \sqrt{3}), (x + \sqrt{3}), (x - 2), (x - 3)\).
4. Multiply the factors to form the polynomial: \((x - \sqrt{3})(x + \sqrt{3})(x - 2)(x - 3)\).
5. Simplify the expression by multiplying the factors: First, use the difference of squares for \((x - \sqrt{3})(x + \sqrt{3})\) to get \(x^2 - 3\), then multiply \((x^2 - 3)(x - 2)(x - 3)\) to get the polynomial.

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 these can be real or complex. The zeros are crucial for constructing the polynomial, as they directly influence its factors and overall shape.

Factoring Polynomials

Factoring a polynomial involves expressing it as a product of its linear factors based on its zeros. For example, if a polynomial has zeros at r1, r2, ..., rn, it can be expressed as f(x) = a(x - r1)(x - r2)...(x - rn), where a is a leading coefficient. This method is essential for constructing the polynomial function from its given zeros.