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Chapter 7: Problem 39

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

Short Answer

Expert verified
\(\frac{2t + 6}{t^2 - 9} = \frac{2}{t - 3}\)

Step by step solution

01

- Factor the Numerator

Identify and factor the numerator. The numerator is \(2t + 6\). Notice that both terms have a common factor of 2. Therefore, factor out 2: \ \[2t + 6 = 2(t + 3)\]
02

- Factor the Denominator

Factor the denominator, which is \(t^2 - 9\). This is a difference of squares, which factors as follows: \ \[t^2 - 9 = (t + 3)(t - 3)\]
03

- Rewrite the Rational Expression

Rewrite the rational expression with the factored numerator and denominator: \ \[\frac{2(t + 3)}{(t + 3)(t - 3)}\]
04

- Simplify the Expression

Cancel the common factor \((t + 3)\) in the numerator and the denominator: \ \[\frac{2 \cancel{(t + 3)}}{\cancel{(t + 3)}(t - 3)} = \frac{2}{t - 3}\]

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is the process of breaking down a polynomial into simpler polynomials that, when multiplied together, give you the original polynomial. This makes solving equations and simplifying expressions easier. For example, to factor the polynomial \(2t + 6\), you look for the greatest common factor (GCF), which is 2 in this case. So, \(2t + 6\) becomes \(2(t + 3)\). Similarly, the polynomial \(t^2 - 9\) can be factored as a difference of squares into \((t + 3)(t - 3)\). This step is crucial in simplifying rational expressions.
Common Factors
Common factors are the numbers or variables that divide each term of a polynomial without leaving a remainder. Identifying common factors helps in simplifying polynomials. For instance, in the numerator \(2t + 6\), both terms share the common factor 2. By factoring out 2, the expression becomes simpler: \(2(t + 3)\). Having common factors in both the numerator and denominator allows us to cancel them out in the next steps, significantly simplifying the rational expression.
Difference of Squares
The difference of squares is a specific type of polynomial where two squared terms are subtracted. It can always be factored into two binomials. The general formula for factoring a difference of squares is: \(a^2 - b^2 = (a + b)(a - b)\). In the example \(t^2 - 9\), we recognize it as a difference of squares because \(t^2\) and 9 are both perfect squares, and it can be factored as \((t + 3)(t - 3)\). This technique is useful for simplifying and solving rational expressions.
Lowest Terms
Simplifying a rational expression to its lowest terms means dividing both the numerator and the denominator by their greatest common factor. This process eliminates unnecessary terms, making the expression easier to work with. After factoring the polynomial \(2t + 6\) to \(2(t + 3)\) and \(t^2 - 9\) to \((t + 3)(t - 3)\), we cancel out the common term \((t + 3)\) in the numerator and denominator. This leaves us with \(\frac{2}{t - 3}\). Always ensure you simplify the expression to its lowest terms for clarity and ease of use.

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Most popular questions from this chapter

Multiply or divide as indicated. $$ \frac{12 x-20}{5 x} \cdot \frac{6}{9 x-15} $$

Multiply or divide as indicated. $$ \frac{a^{3}-b^{3}}{a^{2}-b^{2}} \div \frac{2 a-2 b}{2 a+2 b} $$

Multiply or divide as indicated. $$ \frac{8 x^{3}-27}{2 x^{2}-18} \cdot \frac{2 x+6}{8 x^{2}+12 x+18} $$

Solve each problem. The percent of deaths caused by smoking is modeled by the rational function defined by $$ p(x)=\frac{x-1}{x} $$ where \(x\) is the number of times a smoker is more likely to die of lung cancer than a nonsmoker is. This is called the incidence rate. (Source: Walker, A., Observation and Inference: An Introduction to the Methods of Epidemiology, Epidemiology 91影视 Inc.) For example, \(x=10\) means that a smoker is 10 times more likely than a nonsmoker to die of lung cancer. (a) Find \(p(x)\) if \(x\) is 10 (b) For what values of \(x\) is \(p(x)=80 \% ?\) (Hint: Change \(80 \%\) to a decimal.) (c) Can the incidence rate equal \(0 ?\) Explain.

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