Question

simplify the rational expression \\(\frac{3x^2 - 27x + 54}{3x - 3}\\) in the form \\(q(x) + \frac{r(x)}{b(x)}\\). use the keyboard to write your answer in the box. you may use geogebra cas calculator or any cas calculator. the simplified form of the rational expression \\(\frac{3x^2 - 27x + 54}{3x - 3}\\) is \\(\square\\).

Explanation:

Step1: Factor numerator and denominator

Factor $3x^2-27x+54$: $3(x^2-9x+18)=3(x-3)(x-6)$
Factor $3x-3$: $3(x-1)$
Expression: $\frac{3(x-3)(x-6)}{3(x-1)} = \frac{(x-3)(x-6)}{x-1}$

Step2: Expand numerator

Expand $(x-3)(x-6)$: $x^2-9x+18$
Expression: $\frac{x^2-9x+18}{x-1}$

Step3: Polynomial long division

Divide $x^2-9x+18$ by $x-1$:
First term: $x$ (since $x \cdot (x-1)=x^2-x$)
Subtract: $(x^2-9x+18)-(x^2-x)=-8x+18$
Second term: $-8$ (since $-8 \cdot (x-1)=-8x+8$)
Subtract: $(-8x+18)-(-8x+8)=10$
Expression: $x - 8 + \frac{10}{x-1}$

Answer:

$x - 8 + \frac{10}{x-1}$