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 (1)

Ch. 3 - Polynomial and Rational Functions

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Ch. 3 - Polynomial and Rational Functions

문제 69

4장, 문제 69

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 (1)

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Identify the polynomial function given: \(f(x) = x^3 - 2x^2 - x + 2\).

Recall the Remainder Theorem: When a polynomial \(f(x)\) is divided by \(x - k\), the remainder is \(f(k)\).

To find \(f(1)\), substitute \(x = 1\) into the polynomial: \(f(1) = (1)^3 - 2(1)^2 - (1) + 2\).

Simplify the expression step-by-step: calculate each term and combine them to find the value of \(f(1)\).

The coordinates of the corresponding point on the graph of \(f(x)\) are \((1, f(1))\), where \(f(1)\) is the value found in the previous step.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Evaluating Polynomials

Evaluating a polynomial involves substituting a specific value for the variable x and simplifying the expression to find the corresponding output. For example, to find ƒ(1), substitute x = 1 into the polynomial and compute the result.

Coordinates of Points on a Graph

The coordinates of a point on the graph of a function ƒ(x) are given by (x, ƒ(x)). After finding ƒ(k) using the Remainder Theorem, the point (k, ƒ(k)) represents a point on the polynomial's graph, linking algebraic evaluation to geometric interpretation.