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Chapter 2: Problem 61

State the Remainder Theorem.

Short Answer

Expert verified
The Remainder Theorem states: For a polynomial \( P(x) \) and a number \( c \), when \( P(x) \) is divided by \( (x-c) \), the remainder is \( P(c) \).

Step by step solution

01

Identify the Key Concept

The Remainder Theorem is a rule used in polynomial division in mathematics. It acts as a link between the value of a polynomial's function at a certain point and the remainder of the polynomial's division by the corresponding linear factor.
02

Formulate the Theorem

The Remainder Theorem states: For a polynomial \( P(x) \) and a number \( c \), when \( P(x) \) is divided by \( (x-c) \), the remainder is \( P(c) \). That is, if you substitute \( c \) into the polynomial, you get the same result as if you had divided the polynomial by \( (x-c) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Division
Polynomial division is a mathematical process much like long division with numbers, but it involves polynomials. This procedure helps simplify complex polynomials by breaking them down into simpler parts. When you perform polynomial division, you are dividing a polynomial, referred to as the dividend, by another polynomial, called the divisor.
  • If you're dividing by a linear polynomial like \(x - c\), the resulting quotient can help you understand the behavior of the original polynomial.
  • The division process answers the question of how many times the divisor "fits" into the dividend and what is left over, known as the remainder.
The Remainder Theorem ties directly into polynomial division, providing a quick way to find the remainder when dividing by a linear polynomial. This theorem simplifies the division process by avoiding lengthy division calculations.
Linear Factor
A linear factor is a polynomial of the first degree, which means it is as simple as it gets polynomial-wise. It is typically in the form \(x - c\) where \(c\) is a constant. This particular format is crucial in understanding polynomial behavior and performing polynomial division.
  • The linear factor can be derived from the roots of the polynomial. For instance, if \(c\) is a root of the polynomial, then \(x - c\) is a factor of that polynomial.
  • In polynomial division, dividing a polynomial by a linear factor results in a quotient and, typically, a small remainder. The Remainder Theorem allows us to find this remainder quickly and easily.
Understanding linear factors is crucial when breaking down polynomials and analyzing their roots and overall behavior.
Value of a Polynomial Function
The value of a polynomial function at a given point means substituting that number into the polynomial, resulting in the function's output at that point. For example, if you have a polynomial \(P(x)\) and you need to find \(P(c)\), you simply replace \(x\) with \(c\) in the polynomial expression.
  • Evaluating a polynomial at a specific value provides insight into its behavior and characteristics at that point.
  • By the Remainder Theorem, \(P(c)\) also represents the remainder when dividing the polynomial \(P(x)\) by the linear factor \(x - c\).
This connection between substitution and division is what makes the Remainder Theorem so valuable, simplifying the process of understanding polynomial relationships without needing to perform full division.

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