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Magazine Chapter 7: Problem 29
For exercises 7-32, simplify. $$ \frac{12 r^{5}+60 r^{4}}{r^{4}-r^{3}} \cdot \frac{r^{2}-1}{27 r+135} $$
Short Answer
Expert verified
\( \frac{4(r + 1)}{9} \)
Step by step solution
01
- Factor the numerator of the first fraction
Factor out the greatest common factor (GCF) from the numerator of the first fraction. The GCF of 12r^5 and 60r^4 is 12r^4. Therefore, the numerator becomes: \[12r^4(r + 5)\]
02
- Factor the denominator of the first fraction
Factor out the common factor from the denominator of the first fraction. The GCF of r^4 and -r^3 is r^3. Therefore, the denominator becomes: \[r^3(r - 1)\]
03
- Simplify the first fraction
Simplify the first fraction by canceling out the common factors in the numerator and the denominator: \[\frac{12r^4(r + 5)}{r^3(r - 1)} = 12\frac{r + 5}{r - 1}\]
04
- Factor the numerator of the second fraction
Recognize that the numerator of the second fraction is a difference of squares. \[r^2 - 1 = (r - 1)(r + 1)\]
05
- Factor the denominator of the second fraction
Factor out the GCF from the denominator of the second fraction. The GCF of 27r and 135 is 27. Therefore, the denominator becomes: \[27(r + 5)\]
06
- Simplify and multiply the fractions
Combine the factored expressions and simplify by canceling out any common factors between the numerators and denominators: \[12\frac{r + 5}{r - 1} \cdot \frac{(r - 1)(r + 1)}{27(r + 5)}\] The common factors are (r + 5) and 12/27, which reduces to 4/9:\[\frac{4(r - 1)(r + 1)}{9(r - 1)}\] Now, (r - 1) cancels out from the numerator and denominator: \[\frac{4(r + 1)}{9}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Polynomials
Factoring polynomials is essential when simplifying algebraic expressions. Think of it as breaking down complex expressions into simpler parts. For instance, in our given problem, we factored the numerator of the first fraction, \(12r^5 + 60r^4\), by finding the greatest common factor (GCF). Here, the GCF is \(12r^4\). This allows us to rewrite the expression as \(12r^4(r + 5)\). Factoring like this helps in revealing common factors that can be canceled later on, making the overall simplification process much more straightforward and manageable.
Greatest Common Factor
The Greatest Common Factor (GCF) is the highest factor that divides two or more numbers. Finding the GCF helps in factoring expressions. In our exercise, we identified the GCF for both fractions. For example, the GCF of \(12r^5\) and \(60r^4\) is \(12r^4\). This GCF was factored out to simplify the expression. Likewise, for the second fraction's denominator \(27r + 135\), the GCF is 27. This was factored out to simplify the denominator to \(27(r + 5)\). By recognizing and factoring out the GCF, we simplify expressions greatly and pave the way for canceling out common factors.
Rational Expressions
Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying rational expressions often involves factoring both the numerator and denominator and then canceling common factors. For example, the rational expression in the given exercise consists of two fractions: \( \frac{12r^5 + 60r^4}{r^4 - r^3} \) and \( \frac{r^2 - 1}{27r + 135} \). By factoring each part, we converted it into a simpler form, leading to further cancellations. Simplifying rational expressions typically requires multiple steps but breaking it down systematically makes the process clearer and manageable.
Canceling Common Factors
Once you have factored polynomials and identified common factors, the next step is to cancel these out. Canceling common factors means reducing parts of the fraction that appear in both the numerator and the denominator. For instance, after factoring the expressions in our exercise, we canceled common factors such as \( (r + 5)\) and \( (r - 1)\). Also, we simplified \( \frac{12}{27} \) to \( \frac{4}{9} \). By canceling these, we denoted redundant complexity, resulting in the final, simplified fraction: \( \frac{4(r + 1)}{9} \). This simplification process not only makes the expression easier to interpret but also to work with in further calculations.
