Question

practice: add and subtract rational expressions part i
score: 0.5/6 answered: 0/6
question 1
add \\(\frac{3x^2 - 35x}{3x - 1} + \frac{4x + 10}{3x - 1}\\) and state the sum in simplest form.
the domain is \\(x \
eq\\)

Explanation:

Step1: Add the numerators

Since the denominators are the same ($3x - 1$), we add the numerators: $(3x^2 - 35x) + (4x + 10) = 3x^2 - 35x + 4x + 10 = 3x^2 - 31x + 10$. So the expression becomes $\frac{3x^2 - 31x + 10}{3x - 1}$.

Step2: Factor the numerator

Factor $3x^2 - 31x + 10$. We need two numbers that multiply to $3\times10 = 30$ and add to $-31$. Those numbers are $-30$ and $-1$. So, $3x^2 - 30x - x + 10 = 3x(x - 10) - 1(x - 10) = (3x - 1)(x - 10)$. Now the expression is $\frac{(3x - 1)(x - 10)}{3x - 1}$.

Step3: Simplify the fraction

Cancel out the common factor $(3x - 1)$ (note that $3x - 1
eq0$ so $x
eq\frac{1}{3}$). We get $x - 10$.

Step4: Find the domain

The denominator of the original fractions is $3x - 1$. A fraction is undefined when the denominator is zero, so set $3x - 1 = 0$. Solving for $x$: $3x = 1$ $\Rightarrow$ $x = \frac{1}{3}$. So the domain is $x
eq\frac{1}{3}$.

Answer:

The sum in simplest form is $x - 10$ and the domain is $x
eq\frac{1}{3}$.