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Chapter 7: Problem 54

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

Short Answer

Expert verified
-(m+n)

Step by step solution

01

Recognize the difference of squares

Identify that the numerator is a difference of squares: \[ m^2 - n^2 = (m+n)(m-n) \]
02

Rewrite the numerator

Rewrite the numerator as the product of two binomials: \[ \frac{(m+n)(m-n)}{n-m} \]
03

Simplify the denominator

Recognize that \( n - m \) is the negative of \( m - n \), thus: \[ n - m = -(m - n) \]
04

Simplify the rational expression

Substitute and simplify: \[ \frac{(m+n)(m-n)}{-(m-n)} \]
05

Cancel common factors

Cancel the common factors \( (m-n) \): \[ \frac{m+n}{-1} = -(m+n) \]

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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, recognizing differences of squares is crucial for simplifying expressions. The difference of squares formula, given by \[ a^2 - b^2 = (a+b)(a-b) \], allows us to transform a quadratic expression into a product of two binomials. For example, in the provided exercise, we are given \[ m^2 - n^2 \]. By identifying it as a difference of squares, we can simplify it to \[ (m+n)(m-n) \]. This step is key in breaking down complex polynomial expressions into simpler forms, aiding in further simplification.
factoring
Factoring is the process of breaking down a mathematical expression into simpler components, or factors, whose product is the original expression. For example, consider the expression \[ m^2 - n^2 \]. By recognizing it as a difference of squares, we factor it as \[ (m+n)(m-n) \]. Factoring transforms the expression into a product of two binomials, making it easier to simplify further.
common factors
Identifying and canceling common factors in rational expressions helps simplify them. For instance, in the expression \[ \frac{(m+n)(m-n)}{n-m} \], we notice that \[ (m-n) \] is a factor in both the numerator and the denominator. Recognizing that \[ n-m \] is the negative of \[ m-n \], we rewrite the expression as \[ \frac{(m+n)(m-n)}{-(m-n)} \]. The common factor \[ (m-n) \] cancels out, helping us simplify the expression to \[ - (m+n) \].
negative sign manipulation
Handling negative signs correctly is essential in algebraic simplification. In this exercise, we faced the expression \[ \frac{(m+n)(m-n)}{n-m} \]. Recognizing that \[ n-m \] is the negative of \[ m-n \] is crucial. We express \[ n-m \] as \[ -(m-n) \], allowing us to rewrite the expression as \[ \frac{(m+n)(m-n)}{-(m-n)} \]. This step enables us to cancel common factors properly and simplify the rational expression to \[ - (m+n) \].

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Most popular questions from this chapter

Multiply or divide as indicated. $$ \left(\frac{6 k^{2}-13 k-5}{k^{2}+7 k} \div \frac{2 k-5}{k^{3}+6 k^{2}-7 k}\right) \cdot \frac{k^{2}-5 k+6}{3 k^{2}-8 k-3} $$

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{6 x}{6 x^{2}+5 x y-4 y^{2}}-\frac{2 y}{9 x^{2}-16 y^{2}} $$

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

Multiply or divide as indicated. $$ \left(\frac{2 x^{3}+3 x^{2}-2 x}{3 x-15} \div \frac{2 x^{3}-x^{2}}{x^{2}-3 x-10}\right) \cdot \frac{5 x^{2}-10 x}{3 x^{2}+12 x+12} $$

Multiply or divide as indicated. $$ \frac{(2 x+3)(x-4)}{(x+8)(x-4)} \div \frac{(x-4)(x+2)}{(x-4)(x+8)} $$
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