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Chapter 7: Problem 44

Write each rational expression in lowest terms. $$ \frac{12 x^{2}-4 x-5}{8 x^{2}-6 x-5} $$

Short Answer

Expert verified
The rational expression is already in its lowest terms: \(\frac{(4x+1)(3x-5)}{(4x+5)(2x-1)}\).

Step by step solution

01

- Factor the numerators

Factor the numerator of the rational expression. Identify two numbers that multiply to \texttt{12x\textsuperscript{2}-4x-5} and factor it. Factoring yields: \[12x^{2}-4x-5 = (4x+1)(3x-5)\]
02

- Factor the denominators

Similarly, factor the denominator. Identify two numbers that multiply to \texttt{8x\textsuperscript{2}-6x-5} and factor it. Factoring yields: \[8x^{2}-6x-5 = (4x+5)(2x-1)\]
03

- Simplify the expression

Now, the rational expression in factored form looks like: \(\frac{(4x+1)(3x-5)}{(4x+5)(2x-1)}\). Since there are no common factors between the numerator and the denominator, the expression is already in its lowest terms.

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Key Concepts

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

Intermediate Algebra
Intermediate algebra introduces various fundamental concepts that form a bridge to the more complex topics in advanced algebra. One key area is dealing with rational expressions. Rational expressions are fractions that have polynomials in both the numerator and the denominator. To solve problems involving these expressions, it's essential to understand how to manipulate and simplify them properly. As we work through these problems, we rely on skills from different areas, like factoring polynomials and understanding multiplicative inverses.
Simplifying Rational Expressions
Simplifying rational expressions is like reducing fractions. The goal is to make the expression as simple as possible. We do this by:
1. Factoring the numerator and the denominator.
2. Identifying any common factors.
3. Canceling out these common factors.
For instance, in the given problem, we started with the expression \( \frac{12x^{2}-4x-5}{8x^{2}-6x-5} \). By factoring both the numerator and the denominator, we identified that there were no common factors, which means the expression was already simplified to its lowest terms. It's important to remember that we can only cancel factors, not terms that are added or subtracted.
Factoring Polynomials
Factoring polynomials is a critical skill in algebra. It involves breaking down a polynomial into simpler 'factor' polynomials whose product is the original polynomial. Consider the polynomial \(12x^{2}-4x-5\). To factor it, we look for two numbers that multiply to give the product of the leading coefficient (12) and the constant term (-5), and add to give the middle coefficient (-4). This gives us the factors \(4x+1\) and \(3x-5\). Similarly, for the polynomial \(8x^{2}-6x-5\), we find factors \(4x+5\) and \(2x-1\). Factoring properly allows us to simplify more complex algebraic expressions.
Lowest Terms
Writing expressions in their lowest terms is crucial for clarity and simplicity. An expression is in its lowest terms when the numerator and the denominator have no common factors other than 1. In our exercise, after factoring both the numerator \( (4x+1)(3x-5) \) and the denominator \( (4x+5)(2x-1) \), we found that there were no common factors. This confirms that the expression \( \frac{(4x+1)(3x-5)}{(4x+5)(2x-1)} \) is already in its simplest, or lowest, terms. Always ensure this final check to simplify your algebraic work.

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Most popular questions from this chapter

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{3 a}{a+1}+\frac{2 a}{a-3} $$

Add or subtract as indicated. $$ -\frac{2}{3}+\left(-\frac{4}{7}\right)-\frac{1}{8} $$

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{5}{t^{4} u^{7}}-\frac{3}{t^{5} u^{9}}+\frac{6}{t^{10} u} $$

Multiply or divide as indicated. $$ \frac{64 x^{3}+1}{4 x^{2}-100} \cdot \frac{4 x+20}{64 x^{2}-16 x+4} $$

Solve each problem. The average number of vehicles waiting in line to enter a parking area is modeled by the function defined by $$ w(x)=\frac{x^{2}}{2(1-x)} $$ where \(x\) is a quantity between 0 and 1 known as the traffic intensity. (Source: Mannering, \(\mathrm{F}\), and W. Kilareski, Principles of Highway Engineering and Traffic Control, John Wiley and Sons.) For each traffic intensity, find the average number of vehicles waiting (to the nearest tenth). (a) 0.1 (b) 0.8 (c) 0.9 (d) What happens to waiting time as traffic intensity increases? (IMAGE CANNOT COPY)
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