Question

find the least common denominator of \\(\frac{5x}{3x^2 + 5x - 2}\\) and \\(\frac{5}{3x^2 - 7x + 2}\\).

Explanation:

Step1: Factor first denominator

Factor $3x^2 + 5x - 2$:
We find two numbers that multiply to $3\times(-2)=-6$ and add to $5$, which are $6$ and $-1$.
Split the middle term:
$3x^2 + 6x - x - 2 = 3x(x+2) -1(x+2) = (3x-1)(x+2)$

Step2: Factor second denominator

Factor $3x^2 -7x +2$:
We find two numbers that multiply to $3\times2=6$ and add to $-7$, which are $-6$ and $-1$.
Split the middle term:
$3x^2 -6x -x +2 = 3x(x-2) -1(x-2) = (3x-1)(x-2)$

Step3: Identify LCD

The least common denominator is the product of all unique factors with their highest powers. The unique factors are $(3x-1)$, $(x+2)$, and $(x-2)$.
LCD = $(3x-1)(x+2)(x-2)$
We can also expand this:
$(3x-1)(x^2-4) = 3x^3 -12x -x^2 +4 = 3x^3 -x^2 -12x +4$

Answer:

$(3x-1)(x+2)(x-2)$ or $3x^3 -x^2 -12x +4$