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Chapter 0: Problem 12

Reduce each rational expression to lowest terms. $$ \frac{x^{2}+4 x+4}{x^{2}-4} $$

Short Answer

Expert verified
\( \frac{x + 2}{x - 2} \)

Step by step solution

01

- Factor the numerator

First, factor the numerator. The numerator is a quadratic expression: \[ x^2 + 4x + 4 \] Notice that this can be factored into: \[ (x + 2)(x + 2) = (x + 2)^2 \]
02

- Factor the denominator

Next, factor the denominator. The denominator is also a quadratic expression: \[ x^2 - 4 \] Notice that this can be factored using the difference of squares formula: \[ x^2 - 4 = (x - 2)(x + 2) \]
03

- Write the expression with factored forms

Now, rewrite the given rational expression using the factored forms of the numerator and the denominator: \[ \frac{(x + 2)^2}{(x - 2)(x + 2)} \]
04

- Cancel common factors

Identify and cancel the common factors in the numerator and the denominator. In this case, both the numerator and the denominator have a common factor of \( x + 2 \): \[ \frac{(x + 2) \cdot (x + 2)}{(x - 2) \cdot (x + 2)} \rightarrow \frac{x + 2}{x - 2} \]
05

- Write the final simplified expression

After canceling the common factor, the rational expression simplifies to: \[ \frac{x + 2}{x - 2} \]

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Key Concepts

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

Factoring Quadratics
Quadratic equations are polynomial equations of the form \(ax^2 + bx + c\). Factoring quadratics involves rewriting the equation as a product of two binomials. Let's look at the equation from our exercise: \(x^2 + 4x + 4\). Notice the middle term and the constant. We want two numbers that multiply to give 4 (constant) and add up to 4 (coefficient of \(x\)). These numbers are 2 and 2.
We can rewrite the quadratic as \((x + 2)(x + 2)\) or \((x + 2)^2\). This step is crucial because it lets us simplify the expression later on. Remember, factoring quadratics can be tricky, so always double-check your factors by expanding them back out.
Difference of Squares
A difference of squares is a special type of factoring where you have an expression like \(a^2 - b^2\). This can always be factored as \((a - b)(a + b)\).
Here's the denominator of our exercise: \(x^2 - 4\). Notice how this fits the structure of a difference of squares, where \(x^2\) is \(a^2\) and 4 is \(2^2\). So, we can rewrite it as \((x - 2)(x + 2)\).
Recognizing and factoring difference of squares is essential for simplifying rational expressions.
Canceling Common Factors
Once we've factored both the numerator and the denominator, we need to look for common factors to cancel. From our exercise, the factored form is \( \frac{(x+2)(x+2)}{(x+2)(x-2)} \).
Notice that \(x+2\) appears in both the numerator and the denominator. We can cancel one \(x+2\) from each, simplifying the expression. This leaves us with \( \frac{x+2}{x-2} \).
Canceling common factors simplifies rational expressions. Always ensure the factors you're canceling are common to both the numerator and the denominator.

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Most popular questions from this chapter

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$4(3 x+5)^{1 / 3}(2 x+3)^{3 / 2}+3(3 x+5)^{4 / 3}(2 x+3)^{1 / 2} \quad x \geq-\frac{3}{2}$$

Simplify each expression. Assume that all variables are positive when they appear. $$(5 \sqrt{8})(-3 \sqrt{3})$$

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$\left(x^{2}+4\right)^{4 / 3}+x \cdot \frac{4}{3}\left(x^{2}+4\right)^{1 / 3} \cdot 2 x$$

Simplify each expression. $$100^{3 / 2}$$

Expressions that occur in calculus are given. Write each expression as a single quotient in which only positive exponents and radicals appear. $$\frac{\left(9-x^{2}\right)^{1 / 2}+x^{2}\left(9-x^{2}\right)^{-1 / 2}}{9-x^{2}} \quad-3
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