Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. Zeros of -2, 1, and 0; ƒ(-1)=-1

Verified step by step guidance

1. Identify the zeros of the polynomial function. Since the zeros are -2, 1, and 0, the factors of the polynomial are \((x + 2)\), \((x - 1)\), and \(x\) respectively.
2. Write the general form of the polynomial function as \(f(x) = a \cdot x \cdot (x + 2) \cdot (x - 1)\), where \(a\) is a real number coefficient to be determined.
3. Use the given condition \(f(-1) = -1\) to find the value of \(a\). Substitute \(x = -1\) into the polynomial: \(f(-1) = a \cdot (-1) \cdot (-1 + 2) \cdot (-1 - 1)\).
4. Simplify the expression from the substitution to get an equation in terms of \(a\): \(f(-1) = a \cdot (-1) \cdot 1 \cdot (-2) = 2a\).
5. Set the simplified expression equal to the given value \(-1\) and solve for \(a\): \$2a = -1\(, then find \)a\(. Once \)a\( is found, write the final polynomial function \)f(x)\( by substituting \)a$ back into the general form.

Key Concepts

Polynomial Functions and Degree

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents. The degree of a polynomial is the highest exponent of the variable, which determines the general shape and number of roots of the function. For this problem, the polynomial must be cubic (degree 3).

Zeros of a Polynomial

Zeros (or roots) of a polynomial are the values of x for which the function equals zero. If a polynomial has real coefficients and zeros at -2, 1, and 0, then (x + 2), (x - 1), and x are factors of the polynomial. The polynomial can be expressed as a product of these factors, possibly multiplied by a constant.

Using a Point to Find the Leading Coefficient

Given a point on the polynomial, such as ƒ(-1) = -1, you can substitute x = -1 into the factored form to solve for the unknown leading coefficient. This step ensures the polynomial not only has the correct zeros but also passes through the specified point, fully determining the function.