Simplify $$ \frac{x^{3}-6 x^{2}+9 x}{x^{3}-3 x^{2}} $$

Question

UpStudy AI · Accepted Answer

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:**

$$
\frac{x - 3}{x}
$$
$$ \frac{x - 3}{x} $$

Simplify \( \frac{x^{3}-6 x^{2}+9 x}{x^{3}-3 x^{2}} \)