For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = x² + 5x+6; k = -2

Verified step by step guidance

1. Recall the Remainder Theorem, which states that the remainder when a polynomial ƒ(x) is divided by (x - k) is equal to ƒ(k). This means to find ƒ(k), you simply evaluate the polynomial at x = k.
2. Identify the given polynomial and the value of k. Here, ƒ(x) = x^2 + 5x + 6 and k = -2.
3. Substitute k = -2 into the polynomial: replace every x in the polynomial with -2, so you write ƒ(-2) = (-2)^2 + 5(-2) + 6.
4. Simplify each term step-by-step: calculate (-2)^2, then multiply 5 by -2, and finally add 6.
5. Add all the simplified terms together to find the value of ƒ(-2), which is the remainder when ƒ(x) is divided by (x + 2).

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. Understanding the structure of polynomials, such as ƒ(x) = x² + 5x + 6, is essential for evaluating the function at specific values.

Remainder Theorem

The Remainder Theorem states that when a polynomial ƒ(x) is divided by (x - k), the remainder is equal to ƒ(k). This theorem allows us to find the value of the polynomial at x = k without performing full polynomial division.

Evaluating Polynomials at a Given Value

Evaluating a polynomial at a specific value k involves substituting k into the polynomial expression and simplifying. This process directly gives ƒ(k), which, by the Remainder Theorem, corresponds to the remainder when dividing by (x - k).