Problem 1

Question

The remainder of $p(x)$, when divided by $x-c$, is $p(c)$.

Step-by-Step Solution

Verified

Answer

The remainder when dividing by $ x-c $ is $ p(c) $.

1Step 1: Understand the Question

The problem asks to find the remainder when a polynomial $ p(x) $ is divided by a linear divisor $ x-c $. According to the Remainder Theorem, this remainder is simply $ p(c) $.

2Step 2: Apply the Remainder Theorem

According to the Remainder Theorem, if a polynomial $ p(x) $ is divided by the linear expression $ x-c $, the remainder of this division is $ p(c) $. This means you substitute $ c $ into the polynomial $ p(x) $ to find the remainder.

3Step 3: Example Calculation

For a better understanding, consider a polynomial $ p(x) = 2x^2 - 3x + 5 $ and a value $ c = 2 $. The remainder when $ p(x) $ is divided by $ x-2 $ is $ p(2) = 2(2)^2 - 3(2) + 5 = 8 - 6 + 5 = 7 $. Hence, the remainder is 7.

Key Concepts

Polynomial DivisionLinear DivisorSubstitution MethodRemainder of Polynomial

Polynomial Division

Polynomial division is a crucial concept in understanding how polynomials work, especially when we need to divide by a linear divisor. When you're dividing one polynomial by another, it often mimics long division that we use with regular numbers. But instead of numbers, you're dealing with expressions involving variables.

Let's break this down further: when you divide a polynomial $ p(x) $ by a divisor $ x-c $, the goal is to determine both the quotient $ q(x) $ and the remainder $ r $. In most divisions, we aim to simplify and analyze the relationship between polynomials, leading us towards finding the remainder efficiently using the Remainder Theorem.

Linear Divisor

A linear divisor is simply a polynomial of the form $ x-c $. This is the simplest type of polynomial, containing only a variable to the first power and a constant.

The expression $ x-c $ represents a line that crosses the x-axis at point $ c $.
By using a linear divisor in polynomial division, we're essentially checking how the polynomial behaves when the input is $ c $.

Linear divisors are essential in polynomial division because they simplify the process of determining the remainder. When using the Remainder Theorem, we replace $ x $ with $ c $ to instantly find the remainder, eliminating the need for complex division techniques.

Substitution Method

The substitution method is a technique used to find the remainder of a polynomial division quickly. When dealing with the division of polynomials by a linear divisor, the Remainder Theorem comes into play.

To use this method, you simply substitute $ c $ into your polynomial $ p(x) $. Calculate the value of the polynomial using this input, i.e., find $ p(c) $.

This method is straightforward and saves time compared to traditional polynomial division.
It allows you to quickly evaluate the behavior of the polynomial and determine the remainder efficiently.

By substituting $ c $ into $ p(x) $, we can easily find the remainder without tedious calculations.

Remainder of Polynomial

The remainder of a polynomial, when divided by a linear divisor, is a fundamental concept arising from the Remainder Theorem. This theorem simplifies what could be a complex polynomial division into a straightforward calculation: evaluating the polynomial at a specific point.

The Remainder Theorem states:

If you divide a polynomial $ p(x) $ by $ x-c $, the remainder of this division is $ p(c) $.

For example, if we have a polynomial $ p(x) = 2x^2 - 3x + 5 $ and want to find the remainder when divided by $ x-2 $, we calculate $ p(2) $. The result, $ 7 $, is our remainder. The Remainder Theorem dramatically simplifies finding remainders for polynomials, making it easy to calculate without long division. This approach is both efficient and elegant, providing insights into the behavior of polynomials.

Problem 1

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