Question

find the least common denominator of \\(dfrac{8x}{3x^2 - 11x + 6}\\) and \\(dfrac{3}{3x^2 + 10x - 8}\\).

Explanation:

Step1: Factor the denominators

First, factor \(3x^2 - 11x + 6\). We need two numbers that multiply to \(3\times6 = 18\) and add to \(-11\). Those numbers are \(-9\) and \(-2\). So,

$$
LATEXBLOCK0
$$

Next, factor \(3x^2 + 10x - 8\). We need two numbers that multiply to \(3\times(-8)=-24\) and add to \(10\). Those numbers are \(12\) and \(-2\). So,

$$
LATEXBLOCK1
$$

Step2: Identify the factors for LCD

The denominators are \((3x - 2)(x - 3)\) and \((3x - 2)(x + 4)\). To find the least common denominator (LCD), we take the product of the highest power of each distinct factor. The distinct factors are \((3x - 2)\), \((x - 3)\), and \((x + 4)\).
So, the LCD is \((3x - 2)(x - 3)(x + 4)\). We can also expand this:

$$
LATEXBLOCK2
$$

But usually, the factored form is also acceptable for LCD.

Answer:

The least common denominator is \((3x - 2)(x - 3)(x + 4)\) (or \(3x^3 + x^2 - 38x + 24\) when expanded).