Question

Find the remainder when $x^{7}+7$ is divided by $x-1.$

   Find the remainder when $x^{7}+7$ is divided by $x-1.$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 3, Problem 41

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Solved by Expert Christopher Stanley

07/04/2020

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Step 1: We are given the polynomial $x^{7}+7$ and we want to find the remainder when it is divided by $x-1$. Show more…

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Find the remainder when $x^{7}+7$ is divided by $x-1.$

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Key Concepts

Polynomial Division

Polynomial division is the procedure of dividing one polynomial by another, resulting in a quotient and a remainder. When the divisor is linear, the remainder theorem provides an efficient shortcut by relating the remainder to the value of the polynomial at a certain point.

Remainder Theorem

The remainder theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - a), the remainder of this division is simply f(a). This concept significantly simplifies finding remainders as it avoids complex computations like long division.

Polynomial Evaluation

Polynomial evaluation is the process of computing the value of a polynomial for a given input value. In the context of the remainder theorem, evaluating the polynomial at a specific point directly yields the remainder when divided by a corresponding linear factor.

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