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Chapter 6: Problem 41

Write each rational expression in lowest terms. $$ \frac{t^{2}-9}{3 t+9} $$

Short Answer

Expert verified
\(\frac{t - 3}{3}\)

Step by step solution

01

Identify Common Factors

Determine if the numerator and the denominator have any common factors. The numerator is \(t^2 - 9\) and the denominator is \(3t + 9\).
02

Factor the Numerator

Factor the numerator \(t^2 - 9\) using the difference of squares formula. \(t^2 - 9 = (t - 3)(t + 3)\).
03

Factor the Denominator

Factor the denominator \(3t + 9\) by taking out the greatest common factor, which is 3. \(3t + 9 = 3(t + 3)\).
04

Simplify the Expression

Observe that both the numerator and denominator have a common factor of \(t + 3\). Cancel out the common factor to simplify the expression: \[\frac{(t - 3)(t + 3)}{3(t + 3)} = \frac{t - 3}{3}\].

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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 a key skill in algebra. It involves breaking down a polynomial into simpler components called factors. For example, in the expression \(t^2 - 9\), we can factor it using the difference of squares formula. The polynomial \(t^2 - 9\) can be factored as \((t - 3)(t + 3)\). By identifying and applying the correct factoring technique, complex expressions can be simplified considerably. Always examine the polynomial thoroughly to find the best factoring method that applies.
Common Factors
To simplify rational expressions, identifying common factors is essential. Common factors are terms that appear in both the numerator and the denominator. For instance, consider the original expression \(\frac{t^2 - 9}{3t + 9}\). Upon factoring, the numerator becomes \((t - 3)(t + 3)\) and the denominator becomes \(3(t + 3)\). Here, \(t + 3\) is a common factor. Canceling out these common factors simplifies the expression. Simplifying expressions by identifying and eliminating common factors allows for easier manipulation and understanding of mathematical equations.
Difference of Squares Formula
The difference of squares formula is a specific method used to factor certain types of polynomials. The formula states: \(a^2 - b^2 = (a - b)(a + b)\). This formula is particularly useful when dealing with expressions where both terms are perfect squares, and one is subtracted from the other. For example, in the expression \t^2 - 9\, we recognize that it fits the difference of squares formula where \a = t\ and \b = 3\. Applying the formula gives us \t^2 - 9 = (t - 3)(t + 3)\. This formula simplifies solving and factoring such expressions quickly and correctly.

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Most popular questions from this chapter

Multiply or divide as indicated. $$ \left(\frac{2 x^{3}+3 x^{2}-2 x}{3 x-15} \div \frac{2 x^{3}-x^{2}}{x^{2}-3 x-10}\right) \cdot \frac{5 x^{2}-10 x}{3 x^{2}+12 x+12} $$

Identify the two terms in the numerator and the two terms in the denominator of the rational expression \(\frac{x^{2}+4 x}{x+4},\) and write it in lowest terms.

Add or subtract as indicated. $$\frac{4}{x+3}-\frac{x}{x-3}-\frac{18}{x^{2}-9}$$

Write each rational expression in lowest terms. $$ \frac{p^{2}+q^{2}}{p^{2}-q^{2}} $$

Add or subtract as indicated. $$\frac{3}{x^{2}-5 x+6}-\frac{2}{x^{2}-4 x+4}$$
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