Problem 46 - Precalculus

Question

Explain the mistake that is made. Find the partial-fraction decomposition of \(\frac{3 x^{4}-x-1}{x(x-1)}\) Solution: Write the partial-fraction decomposition form. $$\frac{3 x^{4}-x-1}{x(x-1)}=\frac{A}{x}+\frac{B}{x-1}$$ Multiply both sides by the LCD \(x(x-1)\) $$3 x^{4}-x-1=A(x-1)+B x$$ Eliminate the parentheses and group like terms. $$3 x^{4}-x-1=(A+B) x-A$$ Compare like coefficients. $$A=1, B=-2$$ This is incorrect. What mistake was made?

Step-by-Step Solution

Answer

The mistake was using an incorrect form of partial fractions for the given expression without first using polynomial division.

Step 1: Identify the Problem

We need to find the partial-fraction decomposition of the function \( \frac{3x^4-x-1}{x(x-1)} \). To do this, we assume the expression can be written in the form \( \frac{A}{x} + \frac{B}{x-1} \).

Step 2: Multiply by LCD

Multiply both sides of the equation by the least common denominator, \(x(x-1)\), to clear the denominators. We have:\[3x^4 - x - 1 = A(x-1) + Bx.\]

Step 3: Eliminate Parentheses and Group Terms

Expand the right side of the equation:\[A(x-1) + Bx = Ax - A + Bx.\]Then group like terms:\[(A + B)x - A.\]

Step 4: Compare Like Terms

Now we compare the left side \(3x^4 - x - 1\) with the expanded right side \((A + B)x - A\). The highest degree of \(x\) on the left side is 4, but on the right it's 1, meaning we made a mistake in assuming a partial fraction form suitable for the numerator.

Step 5: Identify the Mistake

The mistake is assuming the numerator has a polynomial degree higher than possible to represent with the given partial-fraction form, which leads to misalignment in the degrees of the terms. Hence, the correct approach is to first perform polynomial long division since \(3x^4 - x - 1\) is of higher degree than \(x(x-1)\).

Step 6: Corrective Action

Perform polynomial long division to express \(\frac{3x^4 - x - 1}{x(x-1)}\) as a polynomial plus a remainder. Only then attempt to decompose the remaining rational expression in partial fractions.

Key Concepts

Rational Expressions

Rational expressions are like fractions but with polynomials. They are expressions of the form \( \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x)\) is not zero. These expressions appear frequently in algebra, making them an important topic to understand.

When dealing with rational expressions, one common task is simplification. This involves factoring both the numerator and the denominator, then reducing by canceling common factors.

Another important aspect is managing operations like addition, subtraction, multiplication, and division. This may require a common denominator, much like with numerical fractions. Here's a quick tip: always look for opportunities to simplify at every step.

• Start by simplifying using factoring.
• Perform arithmetic operations carefully.
• Ensure you understand partial fraction decomposition to break down rational expressions as needed.

For example, in partial fraction decomposition, complex rational expressions are expressed as simpler fractions that add up to the original. Always ensure the degrees are properly managed, which sometimes requires polynomial long division first.

Polynomial Long Division

Polynomial long division is a technique used to divide polynomials, similar to numerical long division. It's particularly useful when the degree of the numerator is greater than or equal to the degree of the denominator.

Here's how it works:

• First, identify the dividend (the polynomial you're dividing) and the divisor (the polynomial you're dividing by).
• Divide the leading term of the dividend by the leading term of the divisor. Place the result above the division symbol.
• Multiply the entire divisor by this term and subtract the result from the dividend. This gives a new dividend.
• Repeat the process until the degree of the new dividend is less than the degree of the divisor.

In the context of partial fraction decomposition, polynomial long division is vital when the numerator has a higher degree than the denominator. Once divided, the result is a polynomial expression plus a smaller rational expression, which can then be decomposed further.

When faced with \( \frac{3x^4-x-1}{x(x-1)} \), polynomial long division helps find the correct form to proceed with decomposition. Always ensure the degree is managed to avoid errors.

Like Terms Comparison

Like terms comparison is a useful method for solving equations involving polynomials. The idea is to ensure that both sides of an equation match in degree and structure by comparing terms individually.

Why is this important? It helps verify the structure and correctness of algebraic manipulations, especially during steps like partial fraction decomposition. Here is how you go about it:

• After multiplying and combining terms, each term's coefficient on one side of the equation should match the corresponding term on the other side.
• If you're decomposing expressions, compare each term of the polynomial individually.

This method was applied incorrectly in the solution originally because the form assumed didn't allow for a complete comparison across all degrees.

For example, assuming \( rac{A}{x} + rac{B}{x-1} \) without considering degrees led to inconsistent structures. Ensuring the highest degree terms are represented properly is critical.