The remainder theorem indicates that when a polynomial ƒ(x) is divided by x-k, the remainder is equal to ƒ(k). Consider the polynomial function ƒ(x) = x³ - 2x² - x+2. Use the remainder theorem to find each of the following. Then determine the coordinates of the corresponding point on the graph of ƒ(x). ƒ (-2)

Verified step by step guidance

1. Identify the polynomial function given: \(f(x) = x^3 - 2x^2 - x + 2\).
2. Recall the Remainder Theorem: When a polynomial \(f(x)\) is divided by \(x - k\), the remainder is \(f(k)\).
3. To find \(f(-2)\), substitute \(x = -2\) into the polynomial: \(f(-2) = (-2)^3 - 2(-2)^2 - (-2) + 2\).
4. Simplify the expression step-by-step: calculate each term separately and then combine them.
5. The value \(f(-2)\) is the remainder when dividing by \(x + 2\), and the corresponding point on the graph is \((-2, f(-2))\).

Key Concepts

Remainder Theorem

The Remainder Theorem states that when a polynomial ƒ(x) is divided by a linear divisor of the form x - k, the remainder of this division is equal to the value of the polynomial evaluated at k, or ƒ(k). This allows for quick calculation of remainders without performing full polynomial division.

Polynomial Evaluation

Polynomial evaluation involves substituting a specific value for the variable x in the polynomial expression and simplifying to find the output. For example, to find ƒ(-2), replace every x in the polynomial with -2 and compute the result.

Coordinates on the Graph of a Function

The coordinates of a point on the graph of a function ƒ(x) are given by (x, ƒ(x)). After evaluating the polynomial at a specific x-value, the resulting pair represents a point on the curve, which helps visualize the function's behavior at that input.