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Chapter 6: Problem 61

Write each rational expression in lowest terms. $$ \frac{(a-3)(x+y)}{(3-a)(x-y)} $$

Short Answer

Expert verified
-\frac{x+y}{x-y}

Step by step solution

01

Identify Common Factors

Observe the rational expression \(\frac{(a-3)(x+y)}{(3-a)(x-y)}\). Notice that the numerator has the factor \(a-3\) and the denominator has the term \(3-a\).
02

Recognize Opposite Factors

Recognize that \(3-a\) is the same as \(-(a-3)\). This means \(3-a = -(a-3)\).
03

Simplify the Factor

Rewrite the rational expression by substituting \(3-a\) with \(-(a-3)\): \(\frac{(a-3)(x+y)}{-(a-3)(x-y)}\).
04

Cancel Common Factors

Now, cancel the common factor \((a-3)\) from the numerator and the denominator. This yields: \(\frac{x+y}{-(x-y)}\).
05

Simplify the Final Expression

Simplify the fraction further by recognizing that \(-(x-y)=-(x-y)= -1 (x-y)\), so \(\frac{x+y}{-(x-y)} = -\frac{x+y}{x-y}\). This is the expression in its lowest terms.

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Key Concepts

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

algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In this exercise, we are working with an algebraic fraction called a rational expression. Understanding algebra helps you simplify and manipulate these expressions.

We use algebraic principles to expand, simplify, and factor expressions. Each step follows rules to ensure the expressions remain equivalent. This is essential when simplifying rational expressions, like turning complex fractions into their simplest form.
rational expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials.

Simplifying rational expressions involves several key steps: identifying common factors, recognizing opposite terms, and canceling out those factors. When you face a rational expression such as \(\frac{(a-3)(x+y)}{(3-a)(x-y)}\), it can look complex, but breaking it down step-by-step as shown in the exercise makes it manageable.

Rational expressions are simplified when the numerator and the denominator have no common factors other than 1. Always rewrite the expression in its lowest terms to make further calculations simpler.
factoring
Factoring is the process of breaking down a polynomial into simpler 'factor' polynomials whose product is the original polynomial.

In the given exercise, we factor the terms in the numerator and the denominator to identify common factors. For example, we see that \((a-3)\) and \((3-a)\) are factors in the numerator and denominator, respectively.

Recognizing that \((3-a) = -(a-3)\) allows us to cancel out common factors effectively. This leads to a simplified expression. Factoring simplifies the process of solving, thereby reducing complexity.

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Most popular questions from this chapter

Write each rational expression in lowest terms. $$ \frac{y^{3}(y-4)}{y^{2}(y-4)} $$

Which rational expression is not equivalent to \(\frac{x-3}{4-x} ?\) A. \(\frac{3-x}{x-4}\) B. \(\frac{x+3}{4+x}\) C. \(-\frac{3-x}{4-x}\) D. \(-\frac{x-3}{x-4}\)

A routine activity such as pumping gasoline can be related to many of the concepts studied in this text. Suppose that premium unleaded costs \$3.75 per gal. 0 gal of gasoline cost \(\$ 0.00\), while 1 gal costs \$3.75. Represent these two pieces of information as ordered pairs of the form (gallons, price).

Add or subtract as indicated. $$\frac{5 x}{x+3}+\frac{x+2}{x}-\frac{6}{x^{2}+3 x}$$

Multiply or divide as indicated. $$ \frac{p^{2}-25}{4 p} \cdot \frac{2}{5-p} $$
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