Problem 59
Question
For exercises 1-66, simplify. $$ \frac{x^{3}-x^{2}-72 x}{x^{4}+5 x^{3}-24 x^{2}} $$
Step-by-Step Solution
Verified Answer
\( \frac{x-9}{x(x-3)} \)
1Step 1: 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 \( x^2 \): \( x^{4} + 5x^{3} - 24x^{2} = x^{2}(x^{2} + 5x - 24) \)
2Step 2: Simplify Common Factors
Cancel the common factor of \( x eq 0 \) from both the numerator and denominator: \( \frac{x(x^{2} - x - 72)}{x^{2}(x^{2} + 5x - 24)} = \frac{x-72}{x(x^{2} + 5x - 24)} \)
3Step 3: 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: \( x^{2} + 5x - 24 = (x + 8)(x - 3) \)
4Step 4: Rewrite Simplified Expression
Replace the factored forms into the fraction: \( \frac{(x-9)(x+8)}{x(x+8)(x-3)} \)
5Step 5: Cancel Common Factors
Cancel the common factor \( x+8 eq 0 \) from the numerator and the denominator: \( \frac{x-9}{x(x-3)} \)
6Step 6: Provide Simplified Expression
The simplified fraction is now: \( \frac{x-9}{x(x-3)} \)
Key Concepts
factoring polynomialscanceling common factorsquadratic expressions
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.
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
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
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
For example, in our numerator quadratic: \begin{align*}(x^{2} - x - 72)\text
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