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Chapter 1: Problem 18

Simplify the rational expression. $$\frac{x^{2}-x-2}{x^{2}-1}$$

Short Answer

Expert verified
\( \frac{x-2}{x-1} \), with \( x \neq -1 \).

Step by step solution

01

Factor the Numerator

The numerator of the rational expression is \( x^2 - x - 2 \). To factor it, we look for two numbers that multiply to \(-2\) and add to \(-1\). These numbers are \(-2\) and \(1\). So, the factored form of the numerator is \((x - 2)(x + 1)\).
02

Factor the Denominator

The denominator is \( x^2 - 1 \), which is a difference of squares. The formula for difference of squares is \( a^2 - b^2 = (a-b)(a+b) \). Here, \( a = x \) and \( b = 1 \), so \( x^2 - 1 \) factors to \((x - 1)(x + 1)\).
03

Simplify by Canceling Common Factors

The rational expression can now be rewritten using the factored forms: \( \frac{(x - 2)(x + 1)}{(x - 1)(x + 1)} \). The common factor in the numerator and the denominator is \((x + 1)\). We can cancel this common factor, simplifying the expression to \( \frac{x - 2}{x - 1} \).
04

State the Simplified Expression

After canceling the common factors, the simplified form of the rational expression is \( \frac{x - 2}{x - 1} \). Additionally, note that \( x eq -1 \) to prevent division by zero in the original expression.

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Key Concepts

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

Factoring Polynomials
Factoring polynomials involves finding expressions that multiply together to give the original polynomial. This is a crucial skill in simplifying rational expressions. Let's take the numerator from our exercise: \( x^2 - x - 2 \). We need to identify two numbers that both add to the coefficient of the middle term, which in this case, is \\(-1\), and multiply to the constant term \\(-2\).
  • The two numbers for \( x^2 - x - 2 \) are \(-2\) and \(1\).
  • When we use these two numbers, we rewrite the expression as \\((x - 2)(x + 1)\).
Factoring correctly is vital as it allows us to simplify expressions more easily later on. Practicing a variety of methods and exercises will strengthen your understanding.
Difference of Squares
The difference of squares is a specific type of polynomial factoring which is very common. It appears in the form \( a^2 - b^2 \) and can be factored into\((a-b)(a+b)\). This method is helpful for simplifying and solving polynomial expressions. In our exercise, the denominator \( x^2 - 1 \) is a classic example of a difference of squares.
  • Here, \( a = x \) and \( b = 1 \).
  • The factored form is \\((x - 1)(x + 1)\).
Recognizing the difference of squares allows for quick simplification and is a skill that can save time and effort in algebra problems.
Canceling Common Factors
Once polynomials have been factored, the next step is to simplify the expression by canceling common factors. Common factors are terms that appear in both the numerator and the denominator, making them eligible to "cancel out". This step reduces the rational expression to its simplest form.
  • For the expression \( \frac{(x - 2)(x + 1)}{(x - 1)(x + 1)} \), the common factor is \\((x + 1)\).
  • Cancelling \( (x + 1) \) results in \\( \frac{x - 2}{x - 1}\).
This simplification is crucial for easier interpretation and computation of expressions. Always remember to ensure that the original factors don't equal zero, to prevent undefined terms.

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Most popular questions from this chapter

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. $$\left(1.062 \times 10^{24}\right)\left(8.61 \times 10^{19}\right)$$

Radicals Simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers. (a) \(\sqrt[5]{x^{3} y^{2}} \sqrt[19]{x^{4} y^{16}}\) (b) \(\frac{\sqrt[3]{8 x^{2}}}{\sqrt{x}}\)

A Family of Equations The equation $$3 x+k-5=k x-k+1$$ is really a family of equations, because for each value of \(k\) we get a different equation with the unknown \(x\). The letter \(k\) is called a parameter for this family. What value should we pick for \(k\) to make the given value of \(x\) a solution of the resulting equation? (a) \(x=0\) (b) \(x=1\) (c) \(x=2\)

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. $$\frac{(73.1)\left(1.6341 \times 10^{28}\right)}{0.0000000019}$$

Gravity If an imaginary line segment is drawn between the centers of the earth and the moon, then the net gravitational force \(F\) acting on an object situated on this line segment is $$F=\frac{-K}{x^{2}}+\frac{0.012 K}{(239-x)^{2}}$$ where \(K>0\) is a constant and \(x\) is the distance of the object from the center of the earth, measured in thousands of miles. How far from the center of the earth is the "dead spot" where no net gravitational force acts upon the object? (Express your answer to the nearest thousand miles.) PICTURE CANT COPY
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