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Chapter 0: Problem 24

Perform the indicated operations and simplify. \(\frac{x^{2}-x-6}{x-3}\)

Short Answer

Expert verified
The simplified expression is \(x + 2\), for \(x \neq 3\).

Step by step solution

01

Factor the Numerator

The first step is to factor the quadratic expression in the numerator: \(x^2 - x - 6\). We need to find two numbers that multiply to \(-6\) (the constant term) and add up to \(-1\) (the coefficient of the middle term). These numbers are \(-3\) and \(2\). Thus, \(x^2 - x - 6\) can be factored as \((x - 3)(x + 2)\).
02

Express the Fraction

Write the fraction with the factored numerator: \(\frac{(x - 3)(x + 2)}{x - 3}\).
03

Simplify by Cancelling Common Factors

Since \(x - 3\) is a common factor in both the numerator and the denominator, we can cancel it out. Be careful to note that this simplification is only valid when \(x eq 3\), as division by zero is undefined. So the expression simplifies to \(x + 2\).

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Key Concepts

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

Factoring Quadratics
When faced with a quadratic expression like \(x^2 - x - 6\), factoring is a key step that simplifies the process of handling algebraic fractions. Quadratics are expressions of the form \(ax^2 + bx + c\). The goal here is to rewrite the quadratic expression as a product of two binomials.

To factor \(x^2 - x - 6\), you need to find two numbers that multiply to the constant term \(-6\) and add to the linear coefficient \(-1\). In this case, \(-3\) and \(2\) are the numbers that satisfy these conditions because:
  • The product of \(-3\) and \(2\) is \(-6\).
  • The sum of \(-3\) and \(2\) is \(-1\).
This means that the quadratic \(x^2 - x - 6\) can be rewritten as \((x - 3)(x + 2)\). Factoring quadratics is crucial for simplifying algebraic fractions and solving quadratic equations.
Simplifying Fractions
Once the quadratic has been factored and the expression is written as a fraction, the next step is to simplify it. For the expression \(\frac{(x - 3)(x + 2)}{x - 3}\), simplifying involves reducing the fraction to its simplest form.

Simplifying a fraction typically means rewriting it so that the numerator and denominator have no common factors, other than 1. In this fraction:
  • The numerator is \((x - 3)(x + 2)\).
  • The denominator is \(x - 3\).
Since \(x - 3\) appears in both the numerator and the denominator, these terms can potentially be canceled out, helping to simplify the fraction. Remember that simplification can only occur when specific conditions are met, such as when the denominator is not zero.
Canceling Common Factors
Canceling common factors is a valuable technique to simplify complex algebraic fractions. Once you've identified the common factors in both the numerator and denominator, you can "cancel" them out. This means they divide each other evenly, simplifying the expression.

For the expression \(\frac{(x - 3)(x + 2)}{x - 3}\), \(x - 3\) is the common factor. By canceling \(x - 3\) from both the numerator and the denominator, you simplify the expression to \(x + 2\). However, it's crucial to indicate the condition that \(x eq 3\).
  • Cancel out common factors only when they exist both in the numerator and denominator.
  • Acknowledge that canceling is valid only if it does not lead to division by zero.
  • Always state any conditions or restrictions, such as values that make the denominator zero.
These steps ensure that the operation is mathematically valid and the expression remains defined.

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Most popular questions from this chapter

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(y=x^{2}(x-1)(x-2)\)

The circular frequency \(v\) of oscillation of a point is given by \(v=\frac{2 \pi}{\text { period }}\). What happens when you add two motions that have the same frequency or period? To investigate, we can graph the functions \(y(t)=2 \sin (\pi t / 5)\) and \(y(t)=\sin (\pi t / 5)+\) \(\cos (\pi t / 5)\) and look for similarities. Armed with this information, we can investigate by graphing the following functions over the interval [-5,5] (a) \(y(t)=3 \sin (\pi t / 5)-5 \cos (\pi t / 5)+2 \sin ((\pi t / 5)-3)\) (b) \(y(t)=3 \cos (\pi t / 5-2)+\cos (\pi t / 5)+\cos ((\pi t / 5)-3)\)

Sketch the graph of each function. (a) \(f(x)=x^{2}-1\) (b) \(g(x)=\frac{x}{x^{2}+1}\) (c) \(h(x)=\left\\{\begin{array}{ll}x^{2} & \text { if } 0 \leq x \leq 2 \\\ 6-x & \text { if } x>2\end{array}\right.\)

Let \(f(x)=\frac{x}{\sqrt{x}-1}\). Find and simplify. (a) \(f\left(\frac{1}{x}\right)\) (b) \(f(f(x))\)

Which of the following functions are odd? Even? Neither even nor odd? (a) \(f(x)=\frac{3 x}{x^{2}+1}\) (b) \(g(x)=|\sin x|+\cos x\) (c) \(h(x)=x^{3}+\sin x\) (d) \(k(x)=\begin{array}{c}x^{2}+1 \\ |x|+x^{4}\end{array}\)
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