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Chapter 7: Problem 54

Divide and, if possible, simplify. $$\frac{x^{2}-1}{x} \div \frac{x+1}{x-1}$$

Short Answer

Expert verified
\(\frac{(x-1)^2}{x}\)

Step by step solution

01

Rewrite the Division as Multiplication

To divide fractions, rewrite the division expression as a multiplication by the reciprocal of the divisor. Here, change the division \(\frac{x^2-1}{x} \div \frac{x+1}{x-1}\) to multiplication: \[\frac{x^2-1}{x} \times \frac{x-1}{x+1}\]
02

Factor the Numerator

Factor the numerator of the first fraction. Notice that \(x^2-1\) is a difference of squares, which can be factored as \( (x+1)(x-1) \). So, \[\frac{(x+1)(x-1)}{x} \times \frac{x-1}{x+1}\]
03

Simplify the Expression

Multiply the fractions: \[\frac{(x+1)(x-1)}{x} \times \frac{x-1}{x+1} = \frac{(x+1)(x-1)(x-1)}{x(x+1)} = \frac{(x-1)^2}{x}\] Cancel out the common factors in the numerator and the denominator: \(x+1\).
04

Final Simplification

After canceling out \(x+1\), the simplified expression is: \[\frac{(x-1)^2}{x}\]

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Key Concepts

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

Division of Fractions
Dividing fractions might seem tricky at first, but it can be made simpler by converting the division into multiplication. This is done by multiplying by the reciprocal of the second fraction (the divisor).
For example, let's look at the original exercise: \ \ \ \[\frac{x^{2}-1}{x} \div\ \frac{x+1}{x-1}\], which turns into \ \ \ \[\frac{x^2-1}{x} \times\ \frac{x-1}{x+1}\].
This change is crucial because it transforms a potentially confusing operation into something more manageable. Once it鈥檚 in this form, you can proceed with multiplication.
Factorization
Factorization involves breaking down a complex expression into simpler parts, or factors, that when multiplied together give back the original expression. In the given exercise, we need to factorize the term \ \(x^2 -1\).

Notice that \ \(x^2 -1\), which is a difference of squares, can be expressed as\ \ \((x+1)(x-1)\).
This gives us: \ \ \[\frac{(x+1)(x-1)}{x} \times\ \frac{x-1}{x+1}\].

Once factored, these simpler expressions are easier to work with when performing further operations like multiplication or cancellation.
Simplification
Simplification is the process of reducing a mathematical expression to its simplest form. After rewriting the division as multiplication and factoring, the next goal is to simplify the expression.

Here鈥檚 our current expression: \ \[\frac{(x+1)(x-1)}{x} \times\ \frac{x-1}{x+1}\].
We multiply the terms to get: \ \[\frac{(x+1)(x-1)(x-1)}{x(x+1)}\].
Now, we can cancel out the common factors: \ \(x+1\) from both the numerator and the denominator.

After canceling, our simplified form is: \ \[\frac{(x-1)^2}{x}\].
Always look for opportunities to cancel out terms to achieve the simplest form of the expression.

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