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Chapter 7: Problem 53

Write each rational expression in lowest terms. $$ \frac{x^{2}-y^{2}}{y-x} $$

Short Answer

Expert verified
-(x + y)

Step by step solution

01

Recognize the Difference of Squares

The numerator is a difference of squares, so rewrite it using the identity: \[ x^2 - y^2 = (x + y)(x - y) \] Therefore, \[ \frac{x^2 - y^2}{y - x} = \frac{(x + y)(x - y)}{y - x} \]
02

Simplify the Denominator

Notice that \( y - x \) is equivalent to \( -(x - y) \). Rewrite the expression: \[ \frac{(x + y)(x - y)}{-(x - y)} = -\frac{(x + y)(x - y)}{x - y} \]
03

Cancel Common Factors

Cancel the common factor \( x - y \) in the numerator and the denominator: \[ -\frac{(x + y)}{1} = -(x + y) \]

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Key Concepts

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

difference of squares
In algebra, the 'difference of squares' is a special factorization pattern. It applies when you have an expression of the form \(a^2 - b^2\).
This can be factored into two binomials: \(a + b\) and \(a - b\).
This principle is useful for simplifying expressions, like in our given exercise.
Here's the identity we use: \[x^2 - y^2 = (x + y)(x - y)\].
By recognizing this pattern, you can break down complicated terms into simpler factors.
In the exercise, we started with \(x^2 - y^2\) and changed it into \( (x + y)(x - y) \).
This helps us see shared factors and simplifies the problem.
factorization
Factorization means breaking down a complex expression into products of simpler factors.
It's like finding the building blocks of an algebraic expression.
In our exercise, we used the factorization of the difference of squares:\[x^2 - y^2 = (x + y)(x - y)\].
Once factored, it became easier to handle and simplify the problem.
When factoring, look for common identities and patterns:
  • Difference of squares
  • Common factors
  • Special binomials

These techniques make solving algebraic expressions less complicated.
canceling common factors
Canceling common factors simplifies a rational expression by removing identical factors from the numerator and denominator.
It's like reducing a fraction to its simplest form.
In our exercise, we noticed that \(x - y\) in the numerator and denominator could cancel each other out:\[\frac{(x + y)(x - y)}{(y - x)}\].
Since \(y - x\) is also \(- (x - y)\), we adjusted and then canceled the common factor: \[- \frac{(x + y)(x - y)}{(x - y)} = -(x + y)\].
This step reduced the expression to a simpler form.
Always ensure you've factored completely before canceling to avoid mistakes.

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Most popular questions from this chapter

Concept Check Write each formula using the "language" of variation. For example, the formula for the circumference of a circle, \(C=2 \pi r,\) can be written as "The circumference of a circle varies directly as the length of its radius." \(d=2 r,\) where \(d\) is the diameter of a circle with radius \(r\)

Simplify. $$ \frac{\frac{3}{5}}{6} $$

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{5}{x+2}+\frac{2}{x^{2}-2 x+4}-\frac{60}{x^{3}+8} $$

Solve each problem. The percent of deaths caused by smoking is modeled by the rational function defined by $$ p(x)=\frac{x-1}{x} $$ where \(x\) is the number of times a smoker is more likely to die of lung cancer than a nonsmoker is. This is called the incidence rate. (Source: Walker, A., Observation and Inference: An Introduction to the Methods of Epidemiology, Epidemiology 91影视 Inc.) For example, \(x=10\) means that a smoker is 10 times more likely than a nonsmoker to die of lung cancer. (a) Find \(p(x)\) if \(x\) is 10 (b) For what values of \(x\) is \(p(x)=80 \% ?\) (Hint: Change \(80 \%\) to a decimal.) (c) Can the incidence rate equal \(0 ?\) Explain.

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{3}{x^{2}-5 x+6}-\frac{2}{x^{2}-4 x+4} $$
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