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Chapter 6: Problem 87

Write each rational expression in lowest terms. $$ \frac{m^{2}-1}{1-m} $$

Short Answer

Expert verified
-(m + 1)

Step by step solution

01

- Factor the Numerator

Consider the numerator, \(m^2 - 1\). This is a difference of squares and can be factored. Recall that \(a^2 - b^2 = (a - b)(a + b)\). Apply this formula: \[ m^2 - 1 = (m - 1)(m + 1) \]
02

- Simplify the Denominator

Look at the denominator, \(1 - m\). Notice that it can be rewritten as \[ 1 - m = -(m - 1) \]. This is equivalent because multiplying by -1 changes the sign of each term in the parenthesis.
03

- Substitute the Factor and Simplify

Substitute the factored numerator and the rewritten denominator into your expression: \[ \frac{(m - 1)(m + 1)}{-(m - 1)} \] Note that \(m - 1\) appears in both the numerator and the denominator. These terms can cancel each other out. Simplify the expression:\[ \frac{m + 1}{-1} = -(m + 1) \].

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Key Concepts

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

difference of squares
The term 'difference of squares' refers to a particular type of polynomial expression.
It takes the form \(a^2 - b^2\). The difference of two squares can be factored using the formula:
\[a^2 - b^2 = (a - b)(a + b)\].
In our original problem, the numerator is \(m^2 - 1\) which fits this pattern, where \(a = m\) and \(b = 1\).
Thus, we can factor it as \((m - 1)(m + 1)\).
Recognizing a difference of squares makes it easier to simplify the expression.
factoring
Factoring is the process of breaking down a complex expression into simpler components, or 'factors,' that when multiplied together give back the original expression.
Factoring helps in simplifying complex rational expressions.
In our example, once we recognize \(m^2 - 1\) as a difference of squares, we can factor it to \((m - 1)(m + 1)\).
Factoring plays a crucial role in simplifying the given expression before we move on to canceling terms.
cancelling terms
Canceling terms is an important step in simplifying rational expressions.
After factoring the numerator and rewriting the denominator, we identify common terms that appear both in the numerator and the denominator.
In our case, the term \(m - 1\) appears in both the numerator and the denominator.
Since any term divided by itself equals 1, these can be canceled out to make the expression simpler.
After canceling, our expression becomes much easier to manage.
reversing signs
Sometimes, you will encounter expressions where terms need to have their signs reversed for simplification.
Take the denominator \(1 - m\) for example. This can be tricky, but we can rewrite it by factoring out \(-1\): \[ 1 - m = -(m - 1) \].
This step changes the sign inside the parenthesis, making it easier to identify and cancel common terms.
Reversing signs correctly is a critical skill to simplify and solve rational expressions efficiently.

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