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Question 1208433: Find the LCM of the given polynomials.
x^2 - x- 12; x^2 - 8x + 16
Let me see.
x^2 - x - 12 = (x - 4)(x + 3)
x^2 - 8x + 16 = (x - 4)(x - 4) = (x - 4)^2
LCM = (x - 4)(x + 3)
You say?
Question:
What about if there are 3 given polynomials?
Sample:
Given x^2 + 4x + 4; x^3 + 2x^2; (x + 2)^3, find the LCM.
Found 2 solutions by greenestamps, math_tutor2020:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
The LCM must contain each factor the greatest number of times it occurs in each polynomial.
x^2-x-12 = (x-4)(x+3)
x^2-8x+16 = (x-4)(x-4) = (x-4)^2
The LCM is (x+3)(x-4)^2.
The process extends easily to 3 or more polynomials:
x^2+4x+4 = (x+2)^2
x^3+2x^2 = x^2(x+2)
(x+2)^3 = (x+2)^3
The LCM is x^2(x+2)^3.
Answer by math_tutor2020(3817)  (Show Source):
You can put this solution on YOUR website!
Let's look at an example involving 3 numbers.
Question: Find the LCM of 12,18, and 80
Solution:
Find the prime factorization of each
12 = 2*2*3
18 = 2*3*3
80 = 2*2*2*2*5
The unique primes are 2,3,5
2 shows up at most 4 times, which means 2^4 is a factor of the LCM.
3 shows up at most twice, so 3^2 is also a factor of the LCM.
5 shows up at most once, so 5^1 is also a factor of the LCM.
The LCM is therefore 2^4*3^2*5^1 = 16*9*5 = 720
Various online LCM calculators can be used to verify.
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With that example in mind, we can extend to polynomials.
x^2-x-12 = (x-4)(x+3)
x^2-8x+16 = (x-4)(x-4)
The unique prime polynomials are x-4 and x+3
(x-4) shows up at most twice while (x+3) shows up at most once
LCM = (x-4)^2*(x+3)
The order of the factor doesn't matter.
WolframAlpha is one of many tools that you can use to verify.
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Now onto 3 polynomials
x^2+4x+4 = (x+2)(x+2)
x^3+2x^2 = x^2(x+2)
(x+2)^3 = (x+2)(x+2)(x+2)
Unique prime polynomials: x, (x+2)
x shows up at most twice, so x^2 is part of the LCM.
(x+2) shows up at most three times, so (x+2)^3 is also part of the LCM.
The LCM is x^2(x+2)^3
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