Find the Least Common multiple of the two expressions: 24x^2-96 & 18x^2+18x-36
To find the least common multiple (LCM) of the given expressions, we need to factorize each expression completely.
First, let's factorize each expression individually:
Expression 1: 24x^2 - 96
Taking out the greatest common factor (GCF), which is 24, we have:
24(x^2 - 4)
Now, we recognize that x^2 - 4 is a difference of squares and can be factored further:
24(x + 2)(x - 2)
Expression 2: 18x^2 + 18x - 36
Taking out the GCF, which is 18, we have:
18(x^2 + x - 2)
Next, we need to factorize the quadratic expression inside the parentheses. We're looking for two numbers that multiply to give -2 (the coefficient of the constant term, -2) and add up to give the coefficient of the middle term, which is 1.
The two numbers that satisfy this condition are 2 and -1. So, we can factorize the quadratic expression as:
18(x + 2)(x - 1)
Now, we have fully factorized each expression. The next step is to find the LCM.
The LCM is the product of the highest powers of all the prime factors involved. Looking at the factorized expressions, we can identify the prime factors as 2, 3, and x.
To find the LCM, we take the highest power of each prime factor:
Highest power of 2: 24(x + 2)(x - 2) has the highest power of 2 as 2^3 = 8.
Highest power of 3: 18(x + 2)(x - 1) has the highest power of 3 as 3^2 = 9.
Highest power of x: The highest power of x is x^2.
So, the LCM of the two expressions is:
8 * 9 * x^2 = 72x^2