lify the following cx $$ \frac{3-3 x}{x^{2}-3 x+2} $$

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UpStudy AI · Accepted Answer

Certainly! Let's simplify the given expression step by step.

**Given Expression:**
$$
\frac{3 - 3x}{x^{2} - 3x + 2}
$$

**Step 1: Factor the Numerator and the Denominator**

1. **Factor the Numerator:**
   $$
   3 - 3x = 3(1 - x) = -3(x - 1)
   $$
   
2. **Factor the Denominator:**
   $$
   x^{2} - 3x + 2
   $$
   We look for two numbers that multiply to $$2$$ and add up to $$-3$$. These numbers are $$-1$$ and $$-2$$.
   $$
   x^{2} - 3x + 2 = (x - 1)(x - 2)
   $$

**Step 2: Rewrite the Expression with Factored Forms**
$$
\frac{3 - 3x}{x^{2} - 3x + 2} = \frac{-3(x - 1)}{(x - 1)(x - 2)}
$$

**Step 3: Cancel Common Factors**
Notice that $$(x - 1)$$ appears in both the numerator and the denominator. As long as $$x \neq 1$$ (to avoid division by zero), we can cancel this term:
$$
\frac{-3\cancel{(x - 1)}}{\cancel{(x - 1)}(x - 2)} = \frac{-3}{x - 2}
$$

**Final Simplified Expression:**
$$
\frac{3 - 3x}{x^{2} - 3x + 2} = \frac{-3}{x - 2}
$$
Alternatively, you might prefer to write it as:
$$
\frac{3}{2 - x}
$$
Both forms are correct and are equivalent to each other.

**Domain Restriction:**
It's important to note that the original expression is undefined when the denominator is zero. Therefore, $$x \neq 1$$ and $$x \neq 2$$.
$$ \frac{3 - 3x}{x^{2} - 3x + 2} = \frac{-3}{x - 2} $$ or equivalently, $$ \frac{3}{2 - x} $$ Both forms are simplified versions of the original expression.

lify the following cx

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