Problem 57 - College Algebra with Corequisite Support

Question

For the following exercises, use a calculator with CAS to answer the questions. Consider \(\frac{x^{k}}{x+1}\) with \(k=1,2,3 .\) What do you expect the result to be if \(k=4\) ?

Step-by-Step Solution

Answer

For \( k=4 \), expect \( \frac{x^4}{x+1} = x^3 - x^2 + x - 1 + \frac{1}{x+1} \).

Step 1: Analyze the Expression for k=1

We start with the expression \( \frac{x^1}{x+1} = \frac{x}{x+1} \). Using a calculator with CAS, we can simplify this expression or discern a pattern by evaluating it for various values of \( x \).

Step 2: Analyze the Expression for k=2

Next, consider the expression \( \frac{x^2}{x+1} \). Simplify it using the calculator. It can be written as \( \frac{x^2}{x+1} = x - 1 + \frac{1}{x+1} \) by performing polynomial division or partial fraction decomposition.

Step 3: Generalize from k=3

Now take \( \frac{x^3}{x+1} \). Simplify it using the calculator to get \( x^2 - x + 1 - \frac{1}{x+1} \). You will notice a pattern emerging as we simplify these expressions for different \( k \) values.

Step 4: Predict the Pattern for k=4

Observe the pattern from previous steps. Notice how each step decreases the power of \( x \) in the polynomial part while eventually leaving a remainder portion as \( \frac{1}{x+1} \) or something similar. Follow this pattern to predict that for \( k=4 \), \( \frac{x^4}{x+1} = x^3 - x^2 + x - 1 + \frac{1}{x+1} \).

Step 5: Verify Using Calculator with CAS

Use the calculator with CAS to verify \( \frac{x^4}{x+1} \). Calculate the result explicitly to ensure the prediction aligns with the computed result. This step confirms the anticipated polynomial and remainder.

Key Concepts

Partial Fraction Decomposition

Partial Fraction Decomposition is a useful technique for simplifying rational expressions. This method involves breaking down a complex rational expression into a sum of simpler fractions, making it easier to understand and integrate or differentiate. It is particularly helpful when dealing with polynomial division.

In the context of the exercise, using partial fraction decomposition, you can transform an expression like \( \frac{x^2}{x+1} \) into a more manageable form. The process typically involves:

• Simplifying the polynomial by dividing the numerator by the denominator.
• Expressing the function as a sum of terms, typically one polynomial term and smaller fractional terms.

This simplification makes finding patterns easier, as seen in several steps where each \( k \) value leads us to a systematic breakdown of the expression.
Overall, partial fraction decomposition is a key concept when working with rational expressions, allowing us to efficiently decompose and simplify them.

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial, as they occur frequently in algebra, calculus, and beyond.

In the given exercise, \( \frac{x^k}{x+1} \) is a typical form of a rational expression. Each time you change the value of \( k \), the complexity of the polynomial in the numerator increases. However, the method of handling these expressions remains rooted in similar techniques, like simplification through polynomial division or partial fraction decomposition.

• Rational expressions can be simplified by dividing the polynomials.
• They exhibit certain patterns based on their structure.
• The denominator provides insights into potential limits or undefined points of the expression.

Understanding these can aid in both simplification tasks and predicting outcomes for higher power values of \( k \). This comprehension helps tackle more advanced mathematics problems involving rational expressions.

Mathematical Patterns

Mathematical Patterns are regularities, repetitions, or predictable behaviors observed within mathematics. Recognizing such patterns is fundamental, as they enable predictions about future outcomes or steps in a mathematical sequence.

The exercise revolves around identifying the pattern when performing polynomial division on \( \frac{x^k}{x+1} \) for various values of \( k \). As seen from the solution steps:

• Each polynomial division shifts the powers of \( x \) downward, forming a new polynomial part.
• There is usually a remainder that can be expressed as a smaller rational expression, \( \frac{1}{x+1} \).
• With each increase in \( k \), the polynomial grows predictably, adding one more degree to the polynomial part.

Learning to identify such patterns not only helps solve this kind of problem more efficiently but reinforces understanding of polynomial behavior in algebraic structures, predicting future terms, and simplifying complex expressions into understandable components.