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. insert step 1: Identify the zeros of the polynomial. The given zeros are -2+\sqrt{5}, -2-\sqrt{5}, -2, and 1.
2. insert step 2: Use the fact that if a polynomial has real coefficients, any non-real zeros must occur in conjugate pairs. Here, -2+\sqrt{5} and -2-\sqrt{5} are conjugate pairs.
3. insert step 3: Write the factors corresponding to each zero. For a zero \alpha, the factor is (x-\alpha). Thus, the factors are (x-(-2+\sqrt{5})), (x-(-2-\sqrt{5})), (x-(-2)), and (x-1).
4. insert step 4: Multiply the conjugate pair factors to form a quadratic factor: [(x+2-\sqrt{5})(x+2+\sqrt{5})]. This simplifies to (x+2)^2 - (\sqrt{5})^2.
5. insert step 5: Multiply all the factors together to form the polynomial: [(x+2)^2 - 5] * (x+2) * (x-1).

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, which 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 x = r1, x = r2, ..., x = 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.