Problem 89 - Algebra for College Students

Question

Perform the indicated operations. $$ \frac{x-1}{2 x^{2}+3 x+1}-\frac{x+1}{2 x^{2}-x-1} $$

Step-by-Step Solution

Answer

\( \frac{-4x}{(2x+1)(x+1)(x-1)} \)

Step 1 - Identify Common Denominators

The given expression is: \( \frac{x-1}{2x^2 + 3x + 1} - \frac{x+1}{2x^2 - x - 1} \). First, identify the denominators: \(2x^2 + 3x + 1\) and \(2x^2 - x - 1\). To combine these fractions, find a common denominator.

Step 2 - Factor the Denominators

Next, factor the denominators if possible. For \(2x^2 + 3x + 1\), it factors to \((2x + 1)(x + 1)\). For \(2x^2 - x - 1\), it factors to \((2x + 1)(x - 1)\).

Step 3 - Determine the Common Denominator

Since \(2x^2 + 3x + 1\) and \(2x^2 - x - 1\) share the factor \(2x + 1\), the common denominator is \((2x+1)(x+1)(x-1)\).

Step 4 - Express Fractions with Common Denominator

Rewrite both fractions with the common denominator: \( \frac{(x-1)(x-1)}{(2x+1)(x+1)(x-1)} - \frac{(x+1)(x+1)}{(2x+1)(x+1)(x-1)} \).

Step 5 - Simplify Numerators

Simplify the numerators: \((x - 1)^2 = x^2 - 2x + 1\) and \((x + 1)^2 = x^2 + 2x + 1\).

Step 6 - Combine the Fractions

Combine the numerators over the common denominator: \( \frac{x^2 - 2x + 1 - (x^2 + 2x + 1)}{(2x+1)(x+1)(x-1)} \).

Step 7 - Simplify the Result

Simplify the combined numerator: \(x^2 - 2x + 1 - x^2 - 2x - 1 = -4x \). Therefore, the expression simplifies to: \( \frac{-4x}{(2x+1)(x+1)(x-1)} \).

Key Concepts

Common Denominator

When working with algebraic fractions, one of the key steps is finding a common denominator. This allows you to combine or compare fractions. For the given problem, the denominators are: \( 2x^2 + 3x + 1 \) and \( 2x^2 - x - 1 \).
To combine these fractions, both fractions must have the same denominator. By identifying a common denominator, we ensure that each fraction is expressed in terms of the same whole. This is crucial when subtracting or adding fractions, as it gives us a common base.

Factoring Polynomials

To find a common denominator, we often need to factor the polynomials in the denominators. Factoring breaks a polynomial down into simpler terms (or factors) that when multiplied together, give the original polynomial.
In this exercise, we factor: \(2x^2 + 3x + 1\) into \((2x + 1)(x + 1)\) and \(2x^2 - x - 1\) into \((2x + 1)(x - 1)\).
Factoring helps reveal common factors, making it easier to identify a common denominator that incorporates all the factors.
This step ensures the fractions are ready to be paired up with a shared denominator for combination or comparison.

Simplifying Expressions

Simplifying expressions involves reducing fractions to their simplest form. In algebra, this can mean combining like terms, canceling common factors, or simplifying the numerator and denominator.
In our exercise, after rewriting the fractions with the common denominator, we simplify the numerators: \((x - 1)^2\) becomes \(x^2 - 2x + 1\) and \((x + 1)^2\) becomes \(x^2 + 2x + 1\).
Then, by subtracting one numerator from the other and simplifying, we ensure the final expression is in its simplest form. The goal of simplification is to make expressions easier to work with and understand.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. When performing operations with rational expressions, such as addition, subtraction, multiplication, or division, it's important to apply rules of arithmetic and algebra.
In the given exercise: \( \frac{x-1}{2x^2 + 3x + 1} - \frac{x+1}{2x^2 - x - 1} \), we treat the numerators and denominators with their polynomial forms.
Operations on rational expressions require us to find common denominators, factor polynomials, and simplify the result.
Mastering these techniques helps in understanding more complex algebraic topics and solving a wide range of algebra problems. The final simplified form of our example is \( \frac{-4x}{(2x+1)(x+1)(x-1)} \).