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Chapter 7: Problem 19

Multiply and, if possible, simplify. $$\frac{x^{2}-3 x-10}{(x-2)^{2}} \cdot \frac{x-2}{x-5}$$

Short Answer

Expert verified
\( \frac{x+2}{x-2} \)

Step by step solution

01

- Factor the Numerator

Factorize the numerator of the first fraction. The numerator is \( x^2 - 3x - 10 \). This can be factored into \((x-5)(x+2)\).
02

- Rewrite the Expression

Rewrite the given expression using the factorized form: \[ \frac{(x-5)(x+2)}{(x-2)^2} \times \frac{x-2}{x-5} \]
03

- Simplify the Fraction

Simplify the expression by cancelling common factors. In \( \frac{(x-5)(x+2)}{(x-2)^2} \times \frac{x-2}{x-5} \), the \( x-5 \) terms cancel out from the numerator and denominator. Simplifying further, we get: \[ \frac{(x+2)}{(x-2)} \times 1 \]
04

- Final Simplification

Finally, the expression simplifies to \( \frac{x+2}{x-2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Polynomials
Factoring polynomials is the process of breaking down a polynomial expression into products of simpler polynomials. This is a fundamental skill in algebra that involves finding two binomials whose product gives the original polynomial. For example, we started with the expression \(x^2 - 3x - 10\). By factoring, we look for two numbers that multiply to the constant term, -10, and add up to the coefficient of the middle term, which is -3. These numbers are -5 and 2. Thus, the factorized form is \((x-5)(x+2)\). This breakdown makes it easier to simplify the expression in the later steps.
Canceling Common Factors
After factoring, the next essential step in simplifying rational expressions is canceling common factors. When we rewrite the expression \(\frac{(x-5)(x+2)}{(x-2)^2} \times \frac{x-2}{x-5}\), we look for terms that appear on both the numerator and the denominator. Here, \(x-5\) and \(x-2\) are common factors. Canceling these out simplifies the expression significantly. Essentially, if a term is both above and below the fraction line, they 'eliminate' each other, leaving their coefficients as 1. This process reduces the complexity and helps in focusing only on the remaining factors.
Simplifying Algebraic Fractions
Simplifying algebraic fractions involves making the fraction as simple as possible. After canceling common factors, the given expression \(\frac{x+2}{x-2} \times 1\) can be simplified by just writing \(\frac{x+2}{x-2}\). This is the simplest form of the given expression. Simplification often involves few steps if each step is tackled methodically: factorization, canceling, and then combining like terms. Remember, simplifying algebraic fractions not only makes them easier to handle but also prepares the expression for potential further operations.

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