Question

adding and subtracting rational expressions quiz
algebra 2 a ae teacher (acc ed) / module 6: rational functions and relations

1. which are equivalent rational expressions for \\(\frac{3x}{x^2+5x-14}\\) and \\(\frac{5}{x^2+10x+21}\\)?

\\(\frac{3x+9}{(x+7)(x-2)(x+3)}\\) and \\(\frac{5x-10}{(x+7)(x-2)(x+3)}
\\(\frac{3x^2+6x}{(x+7)(x-2)(x+3)}\\) and \\(\frac{5x^2-10x}{(x+7)(x-2)(x+3)}
\\(\frac{3x(x+3)}{(x+7)(x-2)(x+3)}\\) and \\(\frac{5(x-2)}{(x+7)(x-2)(x+3)}
\\(\frac{3x^2(x+7)}{(x+7)(x-2)(x+3)}\\) and \\(\frac{5(x+7)}{(x+7)(x-2)(x+3)}\\)

Explanation:

Step1: Factor denominators

Factor $x^2+5x-14=(x+7)(x-2)$ and $x^2+10x+21=(x+7)(x+3)$.

Step2: Find common denominator

Common denominator is $(x+7)(x-2)(x+3)$.

Step3: Adjust first expression

Multiply $\frac{3x}{(x+7)(x-2)}$ by $\frac{x+3}{x+3}$: $\frac{3x(x+3)}{(x+7)(x-2)(x+3)}$.

Step4: Adjust second expression

Multiply $\frac{5}{(x+7)(x+3)}$ by $\frac{x-2}{x-2}$: $\frac{5(x-2)}{(x+7)(x-2)(x+3)}$.

Step5: Match with options

The adjusted expressions match the third option.

Answer:

The third option: $\frac{3x(x+3)}{(x+7)(x-2)(x+3)}$ and $\frac{5(x-2)}{(x+7)(x-2)(x+3)}$