Question

fully simplify the expression below and write your answer as a single fraction.

$\frac{2x^{2}-18}{x^{4}+2x^{3}-3x^{2}}cdot\frac{x^{2}-11x + 10}{x^{2}-13x + 30}$

Explanation:

Step1: Factor the expressions

• Factor $2x^{2}-18 = 2(x^{2}-9)=2(x + 3)(x - 3)$
• Factor $x^{4}+2x^{3}-3x^{2}=x^{2}(x^{2}+2x - 3)=x^{2}(x + 3)(x - 1)$
• Factor $x^{2}-11x + 10=(x-10)(x - 1)$
• Factor $x^{2}-13x + 30=(x - 10)(x - 3)$

Step2: Substitute the factored - forms into the original expression

The original expression $\frac{2x^{2}-18}{x^{4}+2x^{3}-3x^{2}}\cdot\frac{x^{2}-11x + 10}{x^{2}-13x + 30}$ becomes $\frac{2(x + 3)(x - 3)}{x^{2}(x + 3)(x - 1)}\cdot\frac{(x - 10)(x - 1)}{(x - 10)(x - 3)}$

Step3: Cancel out the common factors

Cancel out the common factors $(x + 3)$, $(x - 3)$, $(x - 1)$ and $(x - 10)$ in the numerator and the denominator.
We get $\frac{2}{x^{2}}$

Answer:

$\frac{2}{x^{2}}$