Find the least common multiple of
(2a −1)^2, 1−4a^2, and 8a^3 −12a^2 +6a −1
To find the least common multiple (LCM) of these three expressions, we need to factorize each expression completely.
Let's start with (2a − 1)^2:
(2a − 1)^2 can be expanded as (2a − 1)(2a − 1), which gives us 4a^2 − 4a + 1.
Now, let's factorize 1 − 4a^2:
1 − 4a^2 can be written as (1 − 2a)(1 + 2a).
Lastly, let's factorize 8a^3 − 12a^2 + 6a − 1:
This expression cannot be easily factorized using the rational root theorem or simple factoring techniques.
Now, we can find the LCM. Remember, the LCM is the product of all the unique factors raised to the highest power.
The factors of (2a − 1)^2 are: (2a − 1)(2a − 1)
The factors of (1 − 4a^2) are: (1 − 2a)(1 + 2a)
The factors of (8a^3 − 12a^2 + 6a − 1) cannot be determined.
So, the LCM of (2a − 1)^2, 1 − 4a^2, and 8a^3 − 12a^2 + 6a − 1 is (2a − 1)(2a − 1)(1 − 2a)(1 + 2a).
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To find the least common multiple (LCM) of the given expressions, we need to factorize each expression completely and then find the product of unique factors raised to their highest powers.
Step 1: Factorize each expression completely.
(2a - 1)^2 = (2a - 1)(2a - 1)
1 - 4a^2 = (1 - 2a)(1 + 2a)
8a^3 - 12a^2 + 6a - 1 is already in its simplest form.
Step 2: Identify the unique factors and their highest powers.
The unique factors that appear in the expressions are:
2a - 1, 1 - 2a, 1 + 2a
The highest power for each factor is:
(2a - 1) has a power of 2
(1 - 2a) and (1 + 2a) both have a power of 1
Step 3: Find the LCM by multiplying the unique factors raised to their highest powers.
LCM = (2a - 1)^2 * (1 - 2a) * (1 + 2a)
Therefore, the LCM of (2a - 1)^2, 1 - 4a^2, and 8a^3 - 12a^2 + 6a - 1 is (2a - 1)^2 * (1 - 2a) * (1 + 2a).
Oh, finding the least common multiple, eh? Well, let's put on our detective hats and get to work!
First, let's factorize each of the expressions:
(2a - 1)^2 equals 4a^2 - 4a + 1
1 - 4a^2 is already fully factored
8a^3 - 12a^2 + 6a - 1 can't be factored any further, so we'll leave it as is.
Now, let's identify the unique factors in each expression:
For (2a - 1)^2, we have factors of 2a - 1 and 2a - 1
For 1 - 4a^2, we have factors of 1 - 2a and 1 + 2a
Lastly, 8a^3 - 12a^2 + 6a - 1 has no unique factors.
So, the least common multiple will be the product of all the unique factors raised to their respective highest powers:
(2a - 1)^2 * (1 - 2a) * (1 + 2a) * (8a^3 - 12a^2 + 6a - 1)
Now, if you simplify this expression further, you might get a migraine. So, I'll just give you a friendly piece of advice: use a calculator or computer program to calculate the least common multiple. Trust me, it'll save you from a potential headache!
(2a-1)^2 = (2a-1)(2a-1)
1 - 4a^2 = -(2a-1)(2a+1)
8a^3 - 12a^2 + 6a - 1 = (2a-1)(2a-1)^2
so, what do you think?