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Chapter 7: Problem 61

Divide and, if possible, simplify. \(\frac{a+2}{a-1} \div \frac{3 a+6}{a-5}\)

Short Answer

Expert verified
\(\frac{a-5}{3(a-1)}\)

Step by step solution

01

Rewrite the Division as Multiplication

Rewrite the division problem as a multiplication problem by flipping the second fraction. \[\frac{a+2}{a-1} \times \frac{a-5}{3a+6} \]
02

Factor the Numerator and Denominator

Factor out common terms in the numerators and denominators, if possible. Since there are common factors in the second fraction:\[3a + 6 = 3(a + 2)\] Rewrite the expression: \[\frac{a+2}{a-1} \times \frac{a-5}{3(a+2)}\]
03

Cancel Common Factors

Cancel the common factors from the numerators and denominators. In this case, \(a+2\) is a common factor: \[\frac{a+2}{a-1} \times \frac{a-5}{3(a+2)} = \frac{1}{a-1} \times \frac{a-5}{3} = \frac{a-5}{3(a-1)}\]
04

Write the Simplified Expression

Combine the remaining terms to get the simplified expression: \[\frac{a-5}{3(a-1)}\]

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Key Concepts

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

Division of Fractions
When dividing fractions, the key is to remember a simple rule: flip and multiply. This means you take the reciprocal of the second fraction (flip it) and then multiply by the first fraction.

For the given problem: \(\frac{a+2}{a-1} \div \frac{3a+6}{a-5}\), rewrite the problem as follows: \[ \frac{a+2}{a-1} \times \frac{a-5}{3a+6} \] Now the problem becomes a multiplication of fractions, which makes it easier to handle.
Simplifying Algebraic Expressions
Simplifying algebraic expressions involves reducing the expression to its simplest form.
Sometimes, this requires factoring out common terms from the numerators and denominators before carrying out the operations.
For example, in the numerator and denominator of the second fraction, we have: \[ 3a+6 = 3(a+2) \] After factoring, we rewrite the problem as: \[ \frac{a+2}{a-1} \times \frac{a-5}{3(a+2)} \] Notice how factoring can make it easier to identify and cancel common terms.
Factoring Polynomials
Factoring polynomials is a crucial step in simplifying algebraic expressions.
When you factor, you seek to express the polynomial as a product of simpler terms.
For instance, consider the polynomial: \[ 3a+6 \] This can be factored as: \[ 3(a+2) \] This reveals that both terms share a common factor of 3, making it easier to work with in the context of our problem. By writing polynomials in their factored form, you simplify the process of cancelling common factors.
Canceling Common Factors
Canceling common factors is the final step in simplifying the expression.
After rewriting and factoring, you look for terms that appear in both the numerator and the denominator and cancel them out.
In our example: \[ \frac{a+2}{a-1} \times \frac{a-5}{3(a+2)} \] The term \(a+2\) appears in both the numerator and denominator. By cancelling it, you simplify the expression to: \[ \frac{1}{a-1} \times \frac{a-5}{3} = \frac{a-5}{3(a-1)} \] The final simplified form removes unnecessary terms and makes the expression more manageable.

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