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Chapter 0: Problem 9

Reduce each rational expression to lowest terms. $$ \frac{x^{2}-2 x}{3 x-6} $$

Short Answer

Expert verified
\frac{x}{3}

Step by step solution

01

Factor the numerator

Look at the numerator \(\frac{x^{2}-2 x}{3 x-6}\). Notice that both terms have x in common. Factor out an x.\[x^{2}-2x = x(x-2)\]
02

Factor the denominator

Look at the denominator \(\frac{x^{2}-2 x}{3 x-6}\). Notice that both terms have 3 in common. Factor out a 3.\[3x-6 = 3(x-2)\]
03

Simplify the expression

Now rewrite the rational expression with the factors you found.\[\frac{x(x-2)}{3(x-2)}\]\Since \(x-2\) is a common factor in the numerator and denominator, cancel them out.\[\frac{x(x-2)}{3(x-2)} = \frac{x}{3}\]

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Key Concepts

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

Factoring Polynomials
Polynomials can often look complicated, but breaking them down into simpler factors makes them easier to work with. Think of factoring as finding what you can multiply together to get the original polynomial. For instance, in our exercise, the numerator is \(x^2 - 2x\).

Both terms share a common factor, which in this case is \(x\). By factoring out the common term, the expression becomes \(x(x-2)\).

Similarly, the denominator \(3x - 6\) can be factored out by noticing the common factor \(3\), resulting in \(3(x-2)\).

Factoring allows you to simplify complex expressions, making it easier to see common factors and to perform further mathematical operations.
Simplifying Expressions
Simplifying expressions is all about making them as simple as possible. We do this by canceling out common factors that appear in both the numerator and the denominator.

In the expression \(\frac{x(x-2)}{3(x-2)}\), the \(x-2\) term appears in both the numerator and the denominator. This common factor can be canceled out, significantly simplifying the expression to \(\frac{x}{3}\).

By simplifying, you reduce the number of terms, which makes the expression cleaner and often easier to work with in further calculations. It's like tidying up a messy room; a simpler expression is just easier to handle!
Common Factors
A common factor is a term that divides two or more numbers or expressions exactly. This concept is crucial when simplifying rational expressions.

For example, in the problem we tackled, both the numerator \(x^2 - 2x\) and the denominator \(3x - 6\) have common factors. The numerator shares an \(x\), and the denominator shares a \(3\), but both share the factor \(x-2\).

By identifying and canceling out these common factors, like the \(x-2\) in our problem, we greatly simplify the expression. This technique allows us to solve or manipulate the expression more easily. Always look for common factors when dealing with rational expressions; it's a vital step in simplification.

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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. $$3\left(x^{2}+4\right)^{4 / 3}+x \cdot 4\left(x^{2}+4\right)^{1 / 3} \cdot 2 x$$

Rationalize the numerator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{x}-2}{x-4} x \neq 4$$

Simplify each expression. $$\left(\frac{27}{8}\right)^{2 / 3}$$

Use a calculator to approximate each radical. Round your answer to two decimal places. $$\sqrt[3]{4}$$

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