Question

adding and subtracting rational expressions
complete the following steps to find the lcd and write the sum of the numerators for the given problem:
\\(\frac{2}{x^2 - 3x - 4} + \frac{3}{x^2 - 6x + 8}\\)
factor each denominator.
\\(x^2 - 3x - 4 = \boldsymbol{a}\\) \\(\boldsymbol{(x - 4)(x + 1)\boldsymbol{\downarrow}}\\)
\\(x^2 - 6x + 8 = \boldsymbol{b}\\) \\(\boldsymbol{(x - 4)(x - 2)\boldsymbol{\downarrow}}\\)
the least common denominator is:
\\(\boldsymbol{(x - 4)(x + \boldsymbol{a}\boldsymbol{\downarrow 1})(x - \boldsymbol{b}\boldsymbol{\downarrow 2})}\\)
what is the numerator in the sum?
\\(2(x - 2) + 3(x + 1)\\)
\\(2(x - 2)(x - 4) + 3(x + 1)(x - 4)\\)
\\(2(x + 1)(x - 2)(x - 4) + 3(x + 1)(x - 2)(x - 4)\\)

Explanation:

Step1: Identify the denominators

The first rational expression is \(\frac{2}{x^2 - 3x - 4}\) and the second is \(\frac{3}{x^2 - 6x + 8}\). We factored the denominators: \(x^2 - 3x - 4=(x - 4)(x + 1)\) (let's call this Denominator A) and \(x^2 - 6x + 8=(x - 4)(x - 2)\) (let's call this Denominator B).

Step2: Find the LCD

The least common denominator (LCD) is the product of the highest power of each distinct factor. So, LCD \(=(x - 4)(x + 1)(x - 2)\).

Step3: Rewrite each fraction with LCD

For the first fraction \(\frac{2}{(x - 4)(x + 1)}\), to get the LCD, we multiply numerator and denominator by \((x - 2)\): \(\frac{2(x - 2)}{(x - 4)(x + 1)(x - 2)}\).

For the second fraction \(\frac{3}{(x - 4)(x - 2)}\), to get the LCD, we multiply numerator and denominator by \((x + 1)\): \(\frac{3(x + 1)}{(x - 4)(x + 1)(x - 2)}\).

Step4: Add the numerators

Now, add the numerators (since the denominators are the same now): \(2(x - 2)+3(x + 1)\).

Answer:

\(2(x - 2)+3(x + 1)\) (which corresponds to the first option among the numerator choices)