Find a polynomial function ƒ(x) of least degree with real coefficients having zeros as given. -2+√5, -2-√5, -2, 1

Verified step by step guidance

1. Identify the given zeros of the polynomial: \(-2 + \sqrt{5}\), \(-2 - \sqrt{5}\), \(-2\), and \(1\).
2. Since the polynomial has real coefficients, the conjugate pair \(-2 + \sqrt{5}\) and \(-2 - \sqrt{5}\) will form a quadratic factor. Write this factor as \(\left(x - (-2 + \sqrt{5})\right)\left(x - (-2 - \sqrt{5})\right)\).
3. Simplify the quadratic factor by multiplying the conjugate binomials: \(\left(x + 2 - \sqrt{5}\right)\left(x + 2 + \sqrt{5}\right)\), which can be expressed using the difference of squares formula.
4. Write the factors corresponding to the other zeros as linear factors: \(\left(x - (-2)\right) = (x + 2)\) and \(\left(x - 1\right)\).
5. Form the polynomial function \(f(x)\) by multiplying all factors together: the quadratic factor from step 3 times the linear factors from step 4, resulting in a polynomial of least degree with real coefficients.

Key Concepts

Polynomial Zeros and Factors

Each zero of a polynomial corresponds to a factor of the form (x - zero). To construct a polynomial with given zeros, multiply the factors associated with each zero. For example, a zero at c gives a factor (x - c).

Complex Conjugate Root Theorem

For polynomials with real coefficients, non-real or irrational roots occur in conjugate pairs. Since -2 + √5 and -2 - √5 are conjugates, both must be included to ensure the polynomial has real coefficients.

Constructing the Polynomial of Least Degree

The polynomial of least degree with given zeros is formed by multiplying the linear factors corresponding to each zero exactly once. This ensures the polynomial has the smallest possible degree while including all specified roots.