Find the LCD 1/(x^2)-4/(3x^2+15x)

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

Since ${\displaystyle x^2,3x(x+5)}$ contains both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.

The LCM is the smallest positive number that all of the numbers divide into evenly.

The number ${\displaystyle 1}$ is not a prime number because it only has one positive factor, which is itself.

Since ${\displaystyle 3}$ has no factors besides ${\displaystyle 1}$ and ${\displaystyle 3}$.

The LCM of ${\displaystyle 1,3}$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

The factors for ${\displaystyle x^2}$ are ${\displaystyle x\cdot x}$, which is ${\displaystyle x}$ multiplied by each other ${\displaystyle 2}$ times.

The factor for ${\displaystyle x^1}$ is ${\displaystyle x}$ itself.

The LCM of ${\displaystyle x^2,x^1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

Multiply ${\displaystyle x}$ by ${\displaystyle x}$.

The factor for ${\displaystyle x+5}$ is ${\displaystyle x+5}$ itself.

The LCM of ${\displaystyle x+5}$ is the result of multiplying all factors the greatest number of times they occur in either term.