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Chapter 6: Problem 43

Write each rational expression in lowest terms. $$ \frac{2 t+6}{t^{2}-9} $$

Short Answer

Expert verified
\( \frac{2}{t-3} \)

Step by step solution

01

Factor the Numerator

First, take a look at the numerator, which is \(2t + 6\). Factor out the greatest common factor, which is 2: \[ 2t + 6 = 2(t + 3) \]
02

Factor the Denominator

Next, look at the denominator, \(t^2 - 9\). Recognize this as a difference of squares, which can be factored as: \[ t^2 - 9 = (t + 3)(t - 3) \]
03

Simplify the Expression

Now, rewrite the rational expression with the factored numerator and denominator: \[ \frac{2(t + 3)}{(t + 3)(t - 3)} \] Cancel the common factor \(t + 3\) in the numerator and the denominator: \[ \frac{2 ot{(t + 3)}}{ot{(t + 3)} (t - 3)} = \frac{2}{t - 3} \]

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Key Concepts

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

Factoring Polynomials
Polynomials are expressions that include variables raised to different powers. Factoring polynomials involves breaking them down into simpler pieces called factors. Similar to factoring numbers, where you write a number like 12 as 3\texttimes4, you can factor polynomials. Before factoring a polynomial, identify common factors. For example, in the polynomial \(2t + 6\), the greatest common factor (GCF) is 2. Factoring it out gives you \(2(t + 3)\).
Difference of Squares
A special type of polynomial is called the difference of squares. This form appears as \(a^2 - b^2\). It can always be factored into \((a + b)(a - b)\). For instance, in the problem's denominator, we have \(t^2 - 9\). Seeing that 9 is \(3^2\), we can rewrite it as \(t^2 - 3^2\). Applying the difference of squares rule, we get \((t + 3)(t - 3)\).
Greatest Common Factor
The greatest common factor (GCF) is the largest factor shared by all terms in a polynomial. To find the GCF, you look at each term and identify the highest number that divides them. For example, in the expression \(2t + 6\), the terms are 2t and 6. The largest number that divides both 2 and 6 is 2. So, factor out 2 from the expression, making it \(2(t + 3)\). Recognizing and factoring out the GCF simplifies polynomials and is an essential step in making rational expressions easier to work with.

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Most popular questions from this chapter

Divide. Write each answer in lowest terms. $$ \frac{35 x^{8}}{7 x^{9}} \div \frac{5 x^{5}}{9 x^{6}} $$

Multiply. Write each answer in lowest terms. $$ \frac{12 m^{5}}{-2 m^{2}} \cdot \frac{6 m^{6}}{28 m^{3}} $$

Write the prime factored form of each number. $$ 60 $$

Write four equivalent forms for each rational expression. $$ -\frac{2 x-9}{7 x-1} $$

Simplify by starting at "the bottom" and working upward. $$ 1+\frac{1}{1+\frac{1}{1+1}} $$
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