Problem 18
Question
For exercises \(5-48\), simplify. $$ \frac{u^{2}}{u^{2}+6 u+8}-\frac{4}{u^{2}+6 u+8} $$
Step-by-Step Solution
Verified Answer
\( \frac{u-2}{u+4} \)
1Step 1: Identify Common Denominator
Notice that both terms have the same denominator, which is the quadratic expression \( u^2 + 6u + 8 \). This will allow us to combine the fractions easily.
2Step 2: Combine the Numerators
Since the denominators are the same, subtract the numerators: \( \frac{u^2}{u^2+6u+8} - \frac{4}{u^2+6u+8} = \frac{u^2 - 4}{u^2 + 6u + 8} \).
3Step 3: Factor the Numerator and Denominator
Factor the numerator \( u^2 - 4 \) as \( (u+2)(u-2) \). Factor the denominator \( u^2 + 6u + 8 \) as \( (u+2)(u+4) \). Thus, the expression becomes \( \frac{(u-2)(u+2)}{(u+2)(u+4)} \).
4Step 4: Cancel the Common Factor
Cancel the common factor \( (u+2) \) from the numerator and denominator, which simplifies the expression to \( \frac{u-2}{u+4} \).
Key Concepts
common denominatorfactorizationfraction simplificationcanceling factors
common denominator
When simplifying algebraic fractions, identifying a common denominator is crucial. A common denominator is the same denominator shared by all fractions in the expression.
In this exercise, both fractions already share the same denominator, which is: \(u^2 + 6u + 8\).
This simplifies our work because we don't need to modify the fractions to have a common base.
Just proceed to combine the numerators over this common denominator.
In this exercise, both fractions already share the same denominator, which is: \(u^2 + 6u + 8\).
This simplifies our work because we don't need to modify the fractions to have a common base.
Just proceed to combine the numerators over this common denominator.
factorization
Factorization is breaking down a complex expression into simpler multiplicative components. It often helps in simplifying algebraic expressions.
In our problem, we need to factor both the numerator and the denominator to simplify the fraction further.
First, we look at the numerator: \(u^2 - 4\). This can be factored as \( (u - 2)(u + 2) \).
Next, we factor the denominator: \(u^2 + 6u + 8\), which factors to \( (u + 2)(u + 4) \).
Having factored both numerator and denominator, our fraction is now \( \frac{(u - 2)(u + 2)}{(u + 2)(u + 4)} \).
In our problem, we need to factor both the numerator and the denominator to simplify the fraction further.
First, we look at the numerator: \(u^2 - 4\). This can be factored as \( (u - 2)(u + 2) \).
Next, we factor the denominator: \(u^2 + 6u + 8\), which factors to \( (u + 2)(u + 4) \).
Having factored both numerator and denominator, our fraction is now \( \frac{(u - 2)(u + 2)}{(u + 2)(u + 4)} \).
fraction simplification
Simplifying fractions involves reducing them to their simplest form.
After factoring both the numerator and the denominator in the expression, we look for any common factors to cancel out.
From our problem, we can see that both the numerator and denominator share the common factor: \(u + 2\).
By canceling the \( u + 2 \) factor from both the numerator and the denominator, the fraction becomes simpler.
After factoring both the numerator and the denominator in the expression, we look for any common factors to cancel out.
From our problem, we can see that both the numerator and denominator share the common factor: \(u + 2\).
By canceling the \( u + 2 \) factor from both the numerator and the denominator, the fraction becomes simpler.
canceling factors
Canceling factors is a powerful tool in simplifying algebraic fractions. It involves removing common factors from both the numerator and the denominator.
From our factored fraction \( \frac{(u - 2)(u + 2)}{(u + 2)(u + 4)} \), we see \(u + 2\) is a common factor.
Cancel this common factor to reduce the fraction to: \( \frac{u - 2}{u + 4} \).
Now, the fraction is fully simplified. Always ensure factors you cancel are common to both numerator and denominator.
From our factored fraction \( \frac{(u - 2)(u + 2)}{(u + 2)(u + 4)} \), we see \(u + 2\) is a common factor.
Cancel this common factor to reduce the fraction to: \( \frac{u - 2}{u + 4} \).
Now, the fraction is fully simplified. Always ensure factors you cancel are common to both numerator and denominator.
Other exercises in this chapter
Problem 18
For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{r^{2}+11 r+24}{9 r}}{\frac{r^{2}-64}{27 r^{3}}} $$
View solution Problem 18
For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{4}{15 x+6} ; 12(5 x+2) $$
View solution Problem 18
For exercises 7-32, simplify. $$ \frac{n^{2}-9 n+18}{n^{2}-6 n+9} \cdot \frac{n^{2}-2 n-3}{n^{2}-9 n+18} $$
View solution Problem 19
If the price per share of a company's stock is constant, the relationship of the earnings per share, \(x\), and the financial ratio price to earnings, \(y\), is
View solution