Question

these rational expressions all have the same denominator. write the sum of their numerators over the common denominator.\\(\frac{2x + 1}{x - 3} + \frac{x}{x - 3} + \frac{4}{x - 3}\\)\\(\frac{8x^2 + 4x}{x - 3}\\)\\(\frac{x + 5}{x - 3}\\)\\(\frac{3x + 5}{3(x - 3)}\\)\\(\frac{3x + 5}{x - 3}\\)

Explanation:

Step1: Recall the rule for adding rational expressions with the same denominator.

When adding rational expressions with the same denominator, we use the rule: $\frac{a}{c}+\frac{b}{c}+\frac{d}{c}=\frac{a + b + d}{c}$. Here, the common denominator is $x - 3$, and the numerators are $2x + 1$, $x$, and $4$.

Step2: Add the numerators.

Add the numerators together: $(2x + 1)+x + 4$. Combine like terms: $2x+x+1 + 4=3x + 5$.

Step3: Write the sum over the common denominator.

Using the rule from Step 1, we write the sum of the numerators over the common denominator, so we get $\frac{3x + 5}{x - 3}$.

Answer:

$\frac{3x + 5}{x - 3}$ (which corresponds to the fourth option: $\frac{3x+5}{x - 3}$)