Question
Simplify \( \frac{x^{3}-6 x^{2}+9 x}{x^{3}-3 x^{2}} \)
Ask by Phillips Chavez. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\frac{x - 3}{x}
\]
Solution
To simplify the expression:
\[
\frac{x^{3} - 6x^{2} + 9x}{x^{3} - 3x^{2}}
\]
**Step 1: Factor the Numerator and Denominator**
1. **Factor the Numerator:**
\[
x^{3} - 6x^{2} + 9x = x(x^{2} - 6x + 9) = x(x - 3)^2
\]
2. **Factor the Denominator:**
\[
x^{3} - 3x^{2} = x^{2}(x - 3)
\]
**Step 2: Simplify the Expression**
Now, substitute the factored forms back into the original expression:
\[
\frac{x(x - 3)^2}{x^{2}(x - 3)}
\]
**Cancel Common Factors:**
- **Cancel \(x\):**
\[
\frac{\cancel{x}(x - 3)^2}{x^{\cancel{2}}(x - 3)} = \frac{(x - 3)^2}{x(x - 3)}
\]
- **Cancel one \((x - 3)\):**
\[
\frac{\cancel{(x - 3)}(x - 3)}{x\cancel{(x - 3)}} = \frac{x - 3}{x}
\]
**Final Simplified Form:**
\[
\frac{x - 3}{x}
\]
**Domain Restrictions:**
- \(x \neq 0\) (since the denominator cannot be zero)
- \(x \neq 3\) (original expression is undefined at \(x = 3\))
**Answer:**
\[
\frac{x - 3}{x}
\]
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Simplify this solution Bonus Knowledge
To simplify the expression \( \frac{x^{3}-6 x^{2}+9 x}{x^{3}-3 x^{2}} \), we first factor both the numerator and the denominator. In the numerator: \( x^{3} - 6x^{2} + 9x = x(x^{2} - 6x + 9) = x(x - 3)^{2} \) In the denominator: \( x^{3} - 3x^{2} = x^{2}(x - 3) \) Now, substituting these factored forms into the original expression, we have: \[ \frac{x(x - 3)^{2}}{x^{2}(x - 3)} \] We can cancel one \( x \) from the numerator and one \( (x - 3) \) from the denominator (as long as \( x \neq 0 \) and \( x \neq 3 \)): \[ \frac{(x - 3)}{x} \] Thus, the simplified expression is: \[ \frac{x - 3}{x}, \quad x \neq 0, x \neq 3 \]
