Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, use the Remainder Theorem to find the remainder. $$ \left(x^{4}-1\right) \div(x-4) $$

Step-by-Step Solution

Answer

The remainder is 255.

Step 1: Understand the Remainder Theorem

The Remainder Theorem states that the remainder of the division of a polynomial \(f(x)\) by a linear divisor \((x-a)\) is equal to \(f(a)\). This means that instead of performing polynomial long division, we can find the remainder by evaluating the polynomial \(f(x)\) at \(x = a\).

Step 2: Identify the Polynomial and Divisor

The polynomial given is \(x^4 - 1\), and the divisor is \(x - 4\). Here, \(a\) is 4 since \(x - a = x - 4\).

Step 3: Evaluate the Polynomial at x = 4

To find the remainder, we will substitute \(x = 4\) into the polynomial \(x^4 - 1\).

Step 4: Perform the Substitution and Simplify

Substitute \(4\) into the polynomial: \[ f(4) = 4^4 - 1 \] Calculate \(4^4 = 256\), then subtract 1 from 256: \[ f(4) = 256 - 1 = 255 \]

Step 5: Conclude the Solution

The value obtained from \(f(4) = 255\) is the remainder of the division. According to the Remainder Theorem, the remainder of \( (x^4 - 1) \div (x - 4) \) is 255.

Key Concepts

Polynomial Division

Polynomial division can be a complicated process if approached using traditional long division methods, especially when dealing with large degree polynomials. The goal is to divide one polynomial by another, usually of a lesser degree, to find a quotient and, sometimes, a remainder.

• The dividend is the polynomial that you want to divide. In our example, this is \( x^4 - 1 \).
• The divisor is the polynomial you are dividing by. Here, it's \( x - 4 \).

While traditional division mimics the long division of numbers, it can be simplified with the application of the Remainder Theorem, which provides a shortcut by skipping the entire long division process and directly finding the remainder. Polynomial division isn't applicable only in algebra but also finds uses in calculus and number theory, making it a fundamental aspect of higher mathematics.

Evaluating Polynomials

Evaluating polynomials efficiently is a skill that is greatly beneficial when applying the Remainder Theorem. To evaluate a polynomial at a given value means to substitute that value in place of the polynomial variable.Take the polynomial from our exercise, \( x^4 - 1 \):

• We'll evaluate it by substituting \( x = 4 \) into the polynomial, as the Remainder Theorem dictates.

Here's what the evaluation looks like in practice:\[ f(4) = 4^4 - 1 \]Calculate the power first:

• \( 4^4 = 256 \)

Then subtract 1:

• \( 256 - 1 = 255 \).

The result of this evaluation tells us the remainder of the division \( (x^4 - 1) \div (x - 4) \). Polynomials can be evaluated at any value of \( x \), and the process is often done both numerically and graphically.

Linear Divisor

In polynomial division, a linear divisor refers to a polynomial of the form \( x - a \), where \( a \) is a constant. A linear divisor plays a crucial role in simplifying the process of finding remainders due to its simple structure. The expression \( x - a \) is one of the simplest forms of a polynomial, and its simplicity allows the Remainder Theorem to work effectively. Let's consider how it applies to our example:

• Our linear divisor is \( x - 4 \), identifying \( a \) as 4.

Using the linear divisor in the Remainder Theorem simplifies our work immensely since it transforms the problem into valuating the polynomial at \( a \). Without a linear divisor, completing polynomial division would require following a longer process to potentially find the remainder, reinforcing the powerful simplicity and utility of the Remainder Theorem.