Find the LCM of the following :
1)m^2 - 7m + 12 and m^3 - 2m^2 - 2m -3
2) (r - 1) ^3 and r^4 + r^2 +1 and r^3 - 1
3) 3(p^3 - 1) and (4p^2 - 1 ) and 6(2p^2 + 3p - 2 )
m^2-7m+12 = (m-3)(m-4)
m^3-2m^2-2m-3 = (m-3)(m^2+m+1)
so, LCM = (m-3)(m-4)(m^2+m+1)
Now try the others. What do you get?
can you show the process. how you got
m^3-2m^2-2m-3 = (m-3)(m^2+m+1) . pls
just a little synthetic division trying to find a root. Cubics are not always easy to factor by inspection.
can you show the process pls.
To find the least common multiple (LCM) of given expressions, we need to first factorize each expression and then multiply the highest power of each factor that appears in either expression.
Let's solve each problem step by step:
1) LCM of m^2 - 7m + 12 and m^3 - 2m^2 - 2m - 3:
First, let's factorize each expression:
m^2 - 7m + 12 = (m - 3)(m - 4)
m^3 - 2m^2 - 2m - 3 = (m - 1)(m - 3)(m + 1)
Now, identify the highest power of each factor that appears in either expression:
The factors and their highest powers are:
(m - 1) with power 1
(m - 3) with power 1
(m - 4) with power 1
(m + 1) with power 1
Multiply the highest powers of each factor together to find the LCM:
LCM = (m - 1)(m - 3)(m - 4)(m + 1)
2) LCM of (r - 1)^3, r^4 + r^2 + 1, and r^3 - 1:
First, let's factorize each expression:
(r - 1)^3 = (r - 1)(r - 1)(r - 1)
r^4 + r^2 + 1 = (r^2 + r + 1)(r^2 - r + 1)
r^3 - 1 = (r - 1)(r^2 + r + 1)
Identify the highest power of each factor that appears in either expression:
The factors and their highest powers are:
(r - 1) with power 3
(r^2 + r + 1) with power 1
(r^2 - r + 1) with power 1
Multiply the highest powers of each factor together to find the LCM:
LCM = (r - 1)^3(r^2 + r + 1)(r^2 - r + 1)
3) LCM of 3(p^3 - 1), (4p^2 - 1), and 6(2p^2 + 3p - 2):
First, let's factorize each expression:
3(p^3 - 1) = 3(p - 1)(p^2 + p + 1)
(4p^2 - 1) = (2p + 1)(2p - 1)
6(2p^2 + 3p - 2) = 6(2p - 1)(p + 2)
Identify the highest power of each factor that appears in either expression:
The factors and their highest powers are:
(p - 1) with power 1
(p^2 + p + 1) with power 1
(2p + 1) with power 1
(2p - 1) with power 1
(2p - 1) with power 1
(p + 2) with power 1
Multiply the highest powers of each factor together to find the LCM:
LCM = (p - 1)(p^2 + p + 1)(2p + 1)(2p - 1)(p + 2)(2p - 1)