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Chapter 0: Problem 10

Reduce each rational expression to lowest terms. $$ \frac{15 x^{2}+24 x}{3 x^{2}} $$

Short Answer

Expert verified
\( \frac{5x + 8}{x} \)

Step by step solution

01

Factorize the numerator

First, factorize the numerator. The numerator is \(15x^2 + 24x\). Find the greatest common factor (GCF) of 15 and 24, which is 3. Now, factor out 3 and the common variable term \(x\) from the numerator: \[15x^2 + 24x = 3x(5x + 8)\].
02

Rewrite the rational expression

Rewrite the rational expression using the factored numerator: \[ \frac{15x^2 + 24x}{3x^2} = \frac{3x(5x + 8)}{3x^2} \].
03

Simplify the expression

Cancel common factors in the numerator and the denominator. In this case, \(3x\) in the numerator and \(3x\) in the denominator are common factors: \[ \frac{3x(5x + 8)}{3x^2} = \frac{5x + 8}{x} \]. Make sure \(x eq 0\) to avoid division by zero.

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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 an essential skill in simplifying rational expressions. A polynomial is an expression consisting of variables and coefficients. To reduce a rational expression, start by factoring out the greatest common factor (GCF) from the polynomial.
In the given exercise, the polynomial in the numerator is \(15x^2 + 24x\). The GCF here is 3 and the common variable term is \(x\), so we factorize it as:
  • \(15x^2 + 24x = 3x(5x + 8)\)
This step is crucial as it allows us to identify potential simplifications in the rational expression later on.
Greatest Common Factor
The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. Identifying the GCF is vital in simplifying expressions as it helps in factoring out common elements.
For the polynomial \(15x^2 + 24x\), we note the coefficients are 15 and 24. The GCF of 15 and 24 is 3.
We also observe that each term includes \(x\), so we factor \(x\) too. This gives us:
\(15x^2 + 24x = 3x(5x + 8)\).
By factoring out the GCF, we simplify the process of dividing terms in the next steps.
Simplifying Fractions
Simplifying fractions involves reducing them to their simplest form by canceling common factors. Once we have our rational expression factored, we look for terms that can be canceled out.
Using the factored polynomial, rewrite the fraction: \( \frac{15x^2 + 24x}{3x^2} = \frac{3x(5x + 8)}{3x^2} \).
Here, both the numerator and denominator have a common factor \(3x\). By canceling \(3x\) in both parts, we get:
\(\frac{3x(5x + 8)}{3x^2} = \frac{5x + 8}{x} \).
Always reduce fractions by canceling common factors to reach the simplest form.
Division by Zero
Division by zero is undefined in mathematics because it does not lead to a meaningful result. When simplifying rational expressions, always ensure the denominator is not zero.
In our final simplified form, \( \frac{5x + 8}{x} \), it is crucial to state that \( x eq 0 \).
This condition prevents division by zero and maintains the validity of the expression. Ignoring this rule leads to incorrect and meaningless results.
Therefore, always check the denominator and state any necessary restrictions when dealing with rational expressions to avoid division by zero.

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Most popular questions from this chapter

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