Problem 78
Question
For exercises 39-82, simplify. $$ \frac{b^{2}+7 b-18}{b-2} \div \frac{b+9}{b+4} $$
Step-by-Step Solution
Verified Answer
\(\frac{b+4}{b-2}\)
1Step 1: Factor the numerator of the first fraction
First, factor the quadratic expression in the numerator of the first fraction: \(\frac{b^{2}+7b-18}{b-2}\). To factor \(b^{2}+7b-18\), find two numbers that multiply to -18 and add to 7. These numbers are 9 and -2, so the factored form is \((b+9)(b-2)\):\[\frac{(b+9)(b-2)}{b-2}\].
2Step 2: Simplify the first fraction
Simplify the first fraction by canceling out the common term \((b-2)\). This leaves:\[\frac{(b+9)(b-2)}{b-2} = b+9\].
3Step 3: Set up the division of the fractions
Rewrite the division of the fractions as multiplication by the reciprocal. The original problem is:\[\frac{b+9}{b-2} \times \frac{b+4}{b+9}\].
4Step 4: Cancel out common factors
Notice that \(b+9\) appears in both the numerator of the first fraction and the denominator of the second fraction. Cancel out \(b+9\) to get:\[b+9 \times \frac{b+4}{b+9} = \frac{b+4}{b-2}\].
Key Concepts
Factoring QuadraticsCanceling Common FactorsDivision of Fractions
Factoring Quadratics
When simplifying algebraic expressions, factoring quadratics is often an essential step.
For example, consider the quadratic expression in the numerator \(b^{2} + 7b - 18\).
To factor this, we look for two numbers that multiply to -18 and add to 7. Those numbers are 9 and -2. Thus, the factored form of \(b^{2} + 7b - 18\) is \((b+9)(b-2)\).
Factoring quadratics breaks down complex polynomials into simpler multiplicative components, making them easier to handle in equations.
For example, consider the quadratic expression in the numerator \(b^{2} + 7b - 18\).
To factor this, we look for two numbers that multiply to -18 and add to 7. Those numbers are 9 and -2. Thus, the factored form of \(b^{2} + 7b - 18\) is \((b+9)(b-2)\).
Factoring quadratics breaks down complex polynomials into simpler multiplicative components, making them easier to handle in equations.
Canceling Common Factors
After factoring, the next step often involves simplifying the expression by canceling common factors.
In the given problem, we have the factored form \frac{(b+9)(b-2)}{b-2}\. Notice that the term \(b-2\) appears in both the numerator and the denominator.
By canceling out the common term \(b-2\), we simplify the fraction to \(b+9\).
This process reduces the expression to its simplest form, making it much easier to work with.
In the given problem, we have the factored form \frac{(b+9)(b-2)}{b-2}\. Notice that the term \(b-2\) appears in both the numerator and the denominator.
By canceling out the common term \(b-2\), we simplify the fraction to \(b+9\).
This process reduces the expression to its simplest form, making it much easier to work with.
Division of Fractions
In algebra, division of fractions is simplified by multiplying by the reciprocal.
Given \frac{b+9}{b-2} \div \frac{b+9}{b+4}\, we rewrite this as \frac{b+9}{b-2} \times \frac{b+4}{b+9}\.
This converts the division into multiplication, making it straightforward to handle.
In our example, we notice that \(b+9\) is a common factor in the numerator of the first fraction and the denominator of the second fraction.
Cancelling \(b+9\) from both, we are left with the simplified expression \frac{b+4}{b-2}\.
Understanding this concept helps in performing operations on fractions efficiently.
Given \frac{b+9}{b-2} \div \frac{b+9}{b+4}\, we rewrite this as \frac{b+9}{b-2} \times \frac{b+4}{b+9}\.
This converts the division into multiplication, making it straightforward to handle.
In our example, we notice that \(b+9\) is a common factor in the numerator of the first fraction and the denominator of the second fraction.
Cancelling \(b+9\) from both, we are left with the simplified expression \frac{b+4}{b-2}\.
Understanding this concept helps in performing operations on fractions efficiently.
Other exercises in this chapter
Problem 78
For exercises \(75-78\), one part of simplifying a rational expression is completed. Problem: To simplify \(\frac{\frac{x^{2}-2 x-15}{x^{2}-7 x+10}}{\frac{x^{2}
View solution Problem 78
For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 168 n^{2} w^{6} ; 252 n^{2} w^{2} $$
View solution Problem 78
For exercises 77-86, find any values of the variable for which this expression is undefined. $$ \frac{a-7}{a+2} $$
View solution Problem 79
MRI scans of women with the BRCA1 and BRCA2 genetic mutations that were positive for cancer were wrong five out of six times. (These results are "false positive
View solution