In Exercises 33–68, add or subtract as indicated. x/(x²−2x−24) − x/(x²−7x+6)

Verified step by step guidance

1. Factorize the denominators of both fractions. For the first denominator, \(x^2 - 2x - 24\), find two numbers that multiply to \(-24\) and add to \(-2\). For the second denominator, \(x^2 - 7x + 6\), find two numbers that multiply to \(6\) and add to \(-7\).
2. Rewrite the fractions with the factored denominators. The first fraction becomes \(\frac{x}{(x - 6)(x + 4)}\), and the second fraction becomes \(\frac{x}{(x - 6)(x - 1)}\).
3. Identify the least common denominator (LCD) of the two fractions. The LCD is the product of all unique factors in the denominators: \((x - 6)(x + 4)(x - 1)\).
4. Rewrite each fraction with the LCD as the denominator. Multiply the numerator and denominator of the first fraction by \((x - 1)\), and the numerator and denominator of the second fraction by \((x + 4)\). This gives \(\frac{x(x - 1)}{(x - 6)(x + 4)(x - 1)}\) and \(\frac{x(x + 4)}{(x - 6)(x + 4)(x - 1)}\).
5. Combine the fractions by subtracting the numerators over the common denominator. Simplify the numerator \(x(x - 1) - x(x + 4)\) and leave the denominator as \((x - 6)(x + 4)(x - 1)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial expression as a product of its factors. This is essential for simplifying expressions and finding common denominators in rational expressions. For example, the quadratic expressions in the denominators, x^2 - 2x - 24 and x^2 - 7x + 6, can be factored to facilitate addition or subtraction.

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. When adding or subtracting rational expressions, it is necessary to express each fraction with the same denominator to combine them effectively. In this case, finding the least common denominator (LCD) of the two factored expressions is crucial for performing the operation.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Understanding how to manipulate these expressions, including addition, subtraction, multiplication, and division, is fundamental in algebra. In this problem, the operation involves subtracting two rational expressions, which requires careful handling of the numerators and the common denominator.