Our current assignment is to find the LCM, Least Common Multiple of two polynomials. I think I'm doing it right but I'm not quite sure. For the first question:
2t^2+t-3, 2t^2+5t+3
I have 2t^2+t+3 as the LCM. Is this correct or can somebody step me through the correct method to complete the problem?
2t^2+t-3 = (2t+3)(t-1)
2t^2+5t+3 =(2t+3)(t+1)
Therefore (2t+3)(t-1)(t+1) is a multiple of both polynomials. It is also the Least Common Multiple
(2t+3)(t-1)(t+1) can be written as
(2t+3)(t^2-1)= 2t^3 +3t^2 -2t -3
To find the LCM (Least Common Multiple) of two polynomials, you need to factorize each polynomial completely and then multiply all the factors that appear in either polynomial. Let's go through the steps to find the correct LCM for the given polynomials:
1. Factorize the first polynomial, 2t^2 + t - 3:
We can rewrite it as (2t + 3)(t - 1).
2. Factorize the second polynomial, 2t^2 + 5t + 3:
We can rewrite it as (2t + 3)(t + 1).
3. Identify the common factors:
Notice that both polynomials have the factor (2t + 3). So, this will be a factor of the LCM.
4. Include any remaining unique factors:
For the first polynomial, (t - 1) is a unique factor.
For the second polynomial, (t + 1) is a unique factor.
5. Multiply all the factors together:
LCM = (2t + 3)(t - 1)(t + 1).
So, the correct LCM of 2t^2 + t - 3 and 2t^2 + 5t + 3 is (2t + 3)(t - 1)(t + 1).
You can also expand and simplify this expression to get 2t^3 + 3t^2 - 2t - 3, which is another valid representation of the LCM.