Problem 8 - College Algebra with Corequisite Support

Question

For the following exercises, use the Remainder Theorem to find the remainder. $$ \left(x^{4}+5 x^{3}-4 x-17\right) \div(x+1) $$

Step-by-Step Solution

Answer

The remainder is -17.

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 the same as the value of the polynomial evaluated at \( a \). In this exercise, \( f(x) = x^4 + 5x^3 - 4x - 17 \) and we want to find the remainder when it is divided by \( x+1 \).

Step 2: Identify \( a \) from \( x+1 \)

From the divisor \( x+1 \), set \( x+1 = 0 \) to find \( a \). Solving for \( x \), we get \( x = -1 \). This means \( a = -1 \).

Step 3: Evaluate \( f(a) \)

Substitute \( a = -1 \) into the polynomial \( f(x) = x^4 + 5x^3 - 4x - 17 \) to find the value of the polynomial at \( a = -1 \). Calculate \( f(-1) \):\[-1^4 + 5(-1)^3 - 4(-1) - 17 = 1 - 5 + 4 - 17 = -17 \].

Step 4: Conclude with the Remainder

Since the remainder of the division \( x^4 + 5x^3 - 4x - 17 \) by \( x+1 \) is the value of \( f(-1) \), the remainder is \(-17\).

Key Concepts

Polynomial Division

Polynomial division is a mathematical method used to divide a polynomial by another polynomial, usually of lower degree. This process is similar to long division with numbers. When dividing polynomials, the main aim is to simplify the expression and possibly find the quotient and remainder.

• Think of polynomial division as a method to break down complex expressions into simpler parts.
• The division process continues until the degree of the remaining polynomial is less than the divisor.

For the division to be complete, one should follow steps sequentially ensuring every term in the polynomial is divided by the leading term of the divisor until the end of the expression is reached.

Linear Divisor

A linear divisor is a polynomial of first degree in the divisor, often written in the form of \(x-a\).
A linear divisor simplifies the division process because its structure allows for straightforward identification of the value needed in the Remainder Theorem.

• Example of a linear divisor: \(x+1\).
• By rearranging \(x+1 = 0\), solve to find \(x = -1\), identifying the root for evaluation.

Linear divisors play a critical role in the Remainder and Factor Theorems, providing a simple way to evaluate and understand polynomial behavior at specific points.

Evaluate Polynomial

Evaluating a polynomial involves substituting a specific value for the variable and performing the arithmetic operations to solve.
It's equivalent to replacing every \(x\) variable in the polynomial with the given value and simplifying.

• For example, if \(f(x) = x^4 + 5x^3 - 4x - 17\) and \(a = -1\), replace all \(x\)s with \(-1\) to calculate \(f(-1)\).
• Arithmetic operations will reveal the polynomial's value at point \(x = -1\).

This step is crucial in applying the Remainder Theorem, as it connects the polynomial's value at a point with the division remainder.

Finding Remainder

In polynomial division, particularly when working with linear divisors, the Remainder Theorem provides a quick method for finding the remainder without performing the entire division. Instead, evaluate the polynomial at the divisor's root.
The Remainder Theorem states that the remainder of dividing a polynomial \(f(x)\) by \(x-a\) is \(f(a)\).

• For the given problem, the polynomial is \(f(x) = x^4 + 5x^3 - 4x - 17\) and the divisor is \(x+1\), leading to \(a = -1\).
• By evaluating \(f(-1)\), we found that the remainder is -17.

This method is efficient and saves time over traditional long division, making it a favored approach for quick calculations and checks for linear divisors.