find the LCM of y^2+6y+9 and y^2+y-6
y^3+6y^2+9y=Y*(Y^2+6Y+9)=Y*(Y+3)^2
y^2-6y=Y*(Y-6)
The unique factors are therefore Y, (Y+3)^2 & (Y-6)
Y*(Y+3)^2*(Y-6) is the lcm
y^2 + 6y + 9
= (y+3)^2
y^2 + y - 6
= (y+3)(y-2)
so LCM = (y-2)(y+3)^2
To find the LCM of two expressions, we need to first factorize each expression completely. Let's factorize the given expressions:
Expression 1: y^2 + 6y + 9
This expression is a perfect square trinomial, which means it can be factored as (y + 3)^2.
Expression 2: y^2 + y - 6
This expression can be factored as (y + 3)(y - 2).
Now that we have the factorized form of both expressions, we can find their LCM. To find the LCM, we need to consider the highest power of each factor. If a factor appears in both expressions but with a different power, we take the highest power.
The factorization of the expressions gives us:
Expression 1: (y + 3)^2
Expression 2: (y + 3)(y - 2)
Looking at the factors, we have:
- (y + 3) appears in both expressions with different powers: (y + 3)^2 and (y + 3)(y - 2). We take the higher power, which is (y + 3)^2.
- (y - 2) only appears in Expression 2.
Therefore, the LCM of the two expressions is (y + 3)^2 * (y - 2).
In summary, the LCM of y^2 + 6y + 9 and y^2 + y - 6 is (y + 3)^2 * (y - 2).