How do you find the LCM of #x^8- 12x^7 +36x^6, 3x^2- 108# and #7x+ 42#?

Question

Genesis Allen · Accepted Answer

The LCM is:

#21x^9-126x^8-756x^7+4536x^6#

For this question, it's probably easiest to factor all of the polynomials first: #x^8-12x^7+36x^6 = x^6(x^2-12x+36) = x^6(x-6)^2# #3x^2-108 = 3(x^2-36) = 3(x-6)(x+6)# #7x+42 = 7(x+6)# So the LCM of the scalar factors is that of #1#, #3# and #7#, which is #21# The simplest product of polynomial factors including all of the linear factors we have found, in their multiplicities is: #x^6(x-6)^2(x+6) = x^6(x-6)(x^2-36)# #color(white)(x^6(x-6)^2(x+6)) = x^6(x^3-6x^2-36x+216)# #color(white)(x^6(x-6)^2(x+6)) = x^9-6x^8-36x^7+216x^6# So to get the LCM of the original polynomials, we just need to multiply this by #21#: #21(x^9-6x^8-36x^7+216x^6) = 21x^9-126x^8-756x^7+4536x^6#

Isaiah Anthony · Answer

undefined To find the least common multiple (LCM) of the given expressions, we need to factorize each expression and then determine the highest power of each factor that appears in any of the expressions.

The given expressions are:
1) x^8 - 12x^7 + 36x^6
2) 3x^2 - 108
3) 7x + 42

1) Factoring x^8 - 12x^7 + 36x^6:
x^8 - 12x^7 + 36x^6 = x^6(x^2 - 12x + 36) = x^6(x - 6)^2

2) Factoring 3x^2 - 108:
3x^2 - 108 = 3(x^2 - 36) = 3(x - 6)(x + 6)

3) Factoring 7x + 42:
7x + 42 = 7(x + 6)

Now, we determine the highest power of each factor that appears in any of the expressions:
- The factor x^6 appears in the first expression.
- The factors (x - 6) and (x + 6) appear in the second expression.
- The factor (x + 6) appears in the third expression.

Therefore, the LCM of x^8 - 12x^7 + 36x^6, 3x^2 - 108, and 7x + 42 is:
x^6(x - 6)(x + 6)