For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = x³ - 4x² + 2x+1; k = -1

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). So, to find ƒ(k), we simply evaluate the polynomial at x = k.
2. Substitute k = -1 into the polynomial ƒ(x) = x^3 - 4x^2 + 2x + 1. This means replacing every x in the polynomial with -1.
3. Write the expression after substitution: ƒ(-1) = (-1)^3 - 4(-1)^2 + 2(-1) + 1.
4. Simplify each term step-by-step: calculate (-1)^3, (-1)^2, multiply by coefficients, and then combine all terms.
5. Sum all the simplified terms to find the value of ƒ(-1), which is the remainder when ƒ(x) is divided by (x + 1).

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 terms and degrees, is essential for evaluating and manipulating these functions.

Remainder Theorem

The Remainder Theorem states that the remainder when a polynomial f(x) is divided by (x - k) is equal to f(k). This allows us to find the value of the polynomial at x = k by evaluating f(k) directly, simplifying the process of polynomial division.

Evaluating Polynomials at a Given Value

Evaluating a polynomial at a specific value involves substituting the value into the polynomial and performing arithmetic operations to find the result. This skill is crucial for applying the Remainder Theorem and understanding the behavior of polynomial functions at particular points.