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Chapter 7: Problem 59

For exercises 1-66, simplify. $$ \frac{x^{3}-x^{2}-72 x}{x^{4}+5 x^{3}-24 x^{2}} $$

Short Answer

Expert verified
\( \frac{x-9}{x(x-3)} \)

Step by step solution

01

Identify Common Factors

First, identify any common factors in the numerator and the denominator. In the numerator, factor out an x: \( x^{3} - x^{2} - 72x = x(x^{2} - x - 72) \) Similarly, in the denominator, factor out an \( x^2 \): \( x^{4} + 5x^{3} - 24x^{2} = x^{2}(x^{2} + 5x - 24) \)
02

Simplify Common Factors

Cancel the common factor of \( x eq 0 \) from both the numerator and denominator: \( \frac{x(x^{2} - x - 72)}{x^{2}(x^{2} + 5x - 24)} = \frac{x-72}{x(x^{2} + 5x - 24)} \)
03

Factor Quadratic Expressions

Factor the quadratic expressions in both the numerator and the denominator. The numerator becomes: \( x^{2} - x - 72 = (x - 9)(x + 8) \) The denominator becomes: \( x^{2} + 5x - 24 = (x + 8)(x - 3) \)
04

Rewrite Simplified Expression

Replace the factored forms into the fraction: \( \frac{(x-9)(x+8)}{x(x+8)(x-3)} \)
05

Cancel Common Factors

Cancel the common factor \( x+8 eq 0 \) from the numerator and the denominator: \( \frac{x-9}{x(x-3)} \)
06

Provide Simplified Expression

The simplified fraction is now: \( \frac{x-9}{x(x-3)} \)

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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 a key step in simplifying rational expressions. When you factor a polynomial, you break it down into simpler 'factors' that, when multiplied together, give you the original polynomial.
For example, consider the numerator of our problem, \( x^{3} - x^{2} - 72x \). Notice that each term shares a common factor of \(x\). By factoring out \( x \), we get:

\begin{align*}x^{3} - x^{2} - 72x = x(x^{2} - x - 72)\tag*{}otagewline\text You can further simplify each quadratic expression into two binomials by finding their respective roots.
canceling common factors
Canceling common factors is one of the most effective ways to simplify a rational expression. Once we've factored our polynomials, we often find common factors in both the numerator and the denominator. These common factors can be canceled as long as they aren't zero.
Consider our earlier example after factoring:\begin{align*}\frac{x(x^{2}-x-72)}{x^{2}(x^{2}+5x-24)}\tag{}otag
quadratic expressions
Quadratic expressions often appear in rational expressions. A quadratic expression is typically in the form \( ax^2 + bx + c \). Factoring a quadratic expression often involves finding two numbers that multiply to give you \( ac \), and add to give you \( b \).
For example, in our numerator quadratic: \begin{align*}(x^{2} - x - 72)\text

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Most popular questions from this chapter

For exercises \(35-36, T=\frac{336 \mathrm{gm}}{R}\) represents the relationship of tire diameter, \(T\); gear ratio, \(g\); speed, \(m\); and revolutions of the tire per minute, \(R\). Is the relationship of the given variables a direct variation or an inverse variation? $$ g \text { and } m \text { are constant; the relationship of } T \text { and } R $$

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=3, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=9\).

In 2010, about 2,465,940 Americans died. Find the number of these deaths that were from chronic diseases. Round to the nearest hundred. (Source: www.cdc.gov, Jan. 11, 2012) 7 out of 10 deaths among Americans each year are from chronic diseases. (Source: www.cdc.gov, July 7, 2010)

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