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Chapter 8: Problem 34

Write each rational expression in lowest terms. $$\frac{-12 h^{2}+15}{32 h^{2}-40}$$

Short Answer

Expert verified
The simplified rational expression in lowest terms is \(\frac{3}{8}\).

Step by step solution

01

Identify the Greatest Common Factor (GCF) of the numerator and the denominator

First, we need to find the greatest common factor of the terms in both the numerator and the denominator. For the numerator, the GCF of -12 and 15 is 3. The terms are in the form 3(-4h^2 + 5). For the denominator, the GCF of 32 and 40 is 8. The terms are in the form 8(4h^2 - 5). So, the expression can be rewritten as: \[\frac{3(-4h^{2}+5)}{8(4h^{2}-5)}\]
02

Cancel any common factors

Now, we can see that both the numerator and the denominator have the common factor of \((-4h^2 + 5)\). We can cancel out this factor from both the numerator and the denominator. So the simplified rational expression is: \[\frac{3}{8}\] The rational expression in the lowest term is \(\frac{3}{8}\).

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Key Concepts

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

Greatest Common Factor
Finding the greatest common factor (GCF) is a fundamental step when simplifying rational expressions. The GCF of two or more numbers is the largest number that divides all of them without leaving a remainder. When dealing with polynomials, it's vital to look for common numerical factors as well as variable factors.
  • In the given example, the GCF for the numerator \(-12h^2 + 15\) is 3 because both \(-12\) and \(15\) can be divided by 3.
  • Additionally, for the denominator \(32h^2 - 40\), the GCF is 8, as both \(32\) and \(40\) can be evenly divided by 8.
Finding the GCF allows you to factor expressions, making subsequent steps like canceling common factors much easier.
Numerator and Denominator
In any fraction or rational expression, the top part is called the numerator and the bottom part is called the denominator. In rational expressions, both of these parts can be polynomials.
Understanding the composition of the numerator and denominator is crucial:
  • The numerator in the exercise is \(-12h^2 + 15\), and it represents the part of the expression above the fraction line.
  • The denominator is \(32h^2 - 40\), the part below the fraction line.
Working with these terms requires knowledge of factoring techniques and fluency with algebraic manipulation. By factoring out the GCF from both parts, we can simplify the expression more efficiently.
Rational Expressions Simplification
Simplifying rational expressions involves reducing them to their simplest form by eliminating any common factors. This makes the expressions easier to work with and understand, particularly when solving equations or performing other algebraic operations.
The process of simplification typically includes:
  • Identifying and factoring out the greatest common factor (GCF) from both the numerator and denominator.
  • Rewriting the expression in a form that exposes any common factors.
  • Canceling the common factors to reach the simplest form.
This method helps reveal the essence of the expression, without altering its underlying value.
Cancel Common Factors
Canceling common factors is a key step in simplifying rational expressions. Once these common factors are visible after factoring the numerator and the denominator, they can be removed or "canceled" out of the expression, assuming they are not zero.
  • In our example, after factoring, both the numerator and denominator have the common factor \((-4h^2 + 5)\).
  • By canceling this common factor, the expression simplifies to \(\frac{3}{8}\).
Canceling helps reduce the complexity of the expression and focuses on what's essential, eliminating redundancies and simplifying calculations.

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Most popular questions from this chapter

Solve each equation. $$\frac{8 m-7}{24}=\frac{m}{6}-\frac{7}{8}$$

Solve for the indicated variable. $$Q=\frac{n-k}{5 r} \text { for } r$$

Solve each equation. $$\frac{3}{t^{2}}=\frac{6}{t^{2}+5 t}$$

Solve each equation. $$\frac{8}{3 g^{2}-7 g-6}+\frac{4}{g-3}=\frac{8}{3 g+2}$$

Solve each equation. $$\frac{11}{c+9}=\frac{c}{c-4}-\frac{36-8 c}{c^{2}+5 c-36}$$
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