Problem 15
Question
For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{5 p-5}{4 p+12}}{\frac{10 p+10}{7 p+21}} $$
Step-by-Step Solution
Verified Answer
The simplified form is \( \frac{35(p - 1)}{40(p + 1)} \).
1Step 1 - Simplify the Numerator
First, take the numerator of the main fraction: \ \( \frac{5p - 5}{4p + 12} \ \). Factor the numerator and the denominator separately. The numerator \( 5p - 5 \) can be factored as \( 5(p - 1) \). The denominator \( 4p + 12 \) can be factored as \( 4(p + 3) \). So, the simplified form of the numerator is: \ \ \( \frac{5(p - 1)}{4(p + 3)} \ \).
2Step 2 - Simplify the Denominator
Next, take the denominator of the main fraction: \ \( \frac{10p + 10}{7p + 21} \ \). Again, factor the numerator and the denominator separately. The numerator \( 10p + 10 \) can be factored as \( 10(p + 1) \). The denominator \( 7p + 21 \) can be factored as \( 7(p + 3) \). So, the simplified form of the denominator is: \ \ \( \frac{10(p + 1)}{7(p + 3)} \ \).
3Step 3 - Write as a Multiplication
Now, rewrite the main fraction as a multiplication of the numerator and the reciprocal of the denominator: \[ \frac{\frac{5(p - 1)}{4(p + 3)}}{\frac{10(p + 1)}{7(p + 3)}} = \frac{5(p - 1)}{4(p + 3)} \times \frac{7(p + 3)}{10(p + 1)} \]
4Step 4 - Simplify the Expression
Before multiplying, simplify the expression by canceling out common terms in the numerator and the denominator: \[ \frac{5(p - 1)}{4(p + 3)} \times \frac{7(p + 3)}{10(p + 1)} = \frac{5(p - 1)}{4} \times \frac{7}{10(p + 1)} = \frac{35(p - 1)}{40(p + 1)} \]
Key Concepts
Factoring PolynomialsMultiplying Rational ExpressionsCanceling Common Factors
Factoring Polynomials
Factoring polynomials is essential for simplifying rational expressions. When you factor a polynomial, you are rewriting it as a product of simpler polynomials. For example, take the expression \(4p + 12\). Notice that both terms share a common factor of 4. We can factor this out, resulting in \(4(p + 3)\). This step simplifies working with rational expressions.
Similarly, consider \(5p - 5\). Here, both terms share a common factor of 5. Factoring out the 5, you get \(5(p - 1)\). This makes the expression easier to manage.
Key points to remember:
Similarly, consider \(5p - 5\). Here, both terms share a common factor of 5. Factoring out the 5, you get \(5(p - 1)\). This makes the expression easier to manage.
Key points to remember:
- Identify the greatest common factor (GCF) of the terms.
- Factor out the GCF from the polynomial.
- Rewrite the polynomial as a product of the GCF and the remaining terms.
Multiplying Rational Expressions
Multiplying rational expressions involves a few key steps. First, you need to factor both the numerators and denominators. This ensures that you can simplify the expressions before multiplying them together.
For example, suppose you have two rational expressions: \(\frac{5(p - 1)}{4(p + 3)}\) and \(\frac{10(p + 1)}{7(p + 3)}\).
Once you've factored them, you rewrite the division of these two rational expressions as a multiplication involving the reciprocal of the second rational expression:
\[ \frac{5(p - 1)}{4(p + 3)}\ \times \ \frac{7(p + 3)}{10(p + 1)}\]
Now, you can multiply the numerators and the denominators as follows:
\[\frac{35(p - 1)}{40(p + 1)}\]by reducing the common factors.
For example, suppose you have two rational expressions: \(\frac{5(p - 1)}{4(p + 3)}\) and \(\frac{10(p + 1)}{7(p + 3)}\).
Once you've factored them, you rewrite the division of these two rational expressions as a multiplication involving the reciprocal of the second rational expression:
\[ \frac{5(p - 1)}{4(p + 3)}\ \times \ \frac{7(p + 3)}{10(p + 1)}\]
Now, you can multiply the numerators and the denominators as follows:
- Multiply the numerators: \(5(p - 1) \times 7(p + 3)\)
- Multiply the denominators: \(4(p + 3) \times 10(p + 1)\)
\[\frac{35(p - 1)}{40(p + 1)}\]by reducing the common factors.
Canceling Common Factors
To further simplify rational expressions, you often need to cancel common factors in the numerator and the denominator. This step ensures that the expression is in its simplest form.
Consider our example: \(\frac{5(p - 1)}{4(p + 3)} \times \ \frac{7(p + 3)}{10(p + 1)}\). Before multiplying, you will notice that \(p + 3\) appears in both the numerator and the denominator.
You can cancel these common factors, leading to a simplified version of the expression:
\[\frac{5(p - 1)}{4} \times \ \frac{7}{10(p + 1)}\]
Canceling common factors is crucial for expressing an answer clearly and succinctly.
Consider our example: \(\frac{5(p - 1)}{4(p + 3)} \times \ \frac{7(p + 3)}{10(p + 1)}\). Before multiplying, you will notice that \(p + 3\) appears in both the numerator and the denominator.
You can cancel these common factors, leading to a simplified version of the expression:
\[\frac{5(p - 1)}{4} \times \ \frac{7}{10(p + 1)}\]
- Simplify by multiplying the remaining factors.
- Simplify the expression further if possible.
Canceling common factors is crucial for expressing an answer clearly and succinctly.
Other exercises in this chapter
Problem 14
For exercises 7-32, simplify. $$ \left(\frac{r^{2}}{r^{2}+2 r}\right)\left(\frac{r^{2}-4}{r}\right) $$
View solution Problem 15
For a fixed number of windows, the number of windows washed per hour, \(x\), and the number of hours it takes to wash the windows, \(y\), is an inverse variatio
View solution Problem 15
For exercises 13-24, rewrite each expression as an equivalent expression with the given denominator. $$ \frac{2}{9 a} ; 27 a^{2} b $$
View solution Problem 15
For exercises \(5-48\), simplify. $$ \frac{r^{2}-12 r}{r+2}-\frac{28}{r+2} $$
View solution