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Math  /  Algebra

QuestionDetermine the LCM of the given polynomials Leave your answer in factored form. x2+12x+36 and x2+9x+18x^{2}+12 x+36 \text { and } x^{2}+9 x+18

Studdy Solution

STEP 1

1. We need to find the Least Common Multiple (LCM) of two polynomials.
2. The LCM of polynomials is found by taking the highest power of each factor that appears in the factored forms of the polynomials.

STEP 2

1. Factor each polynomial completely.
2. Identify the unique factors from both polynomials.
3. Determine the LCM by taking the highest power of each factor.

STEP 3

Factor the first polynomial x2+12x+36x^2 + 12x + 36.
Recognize it as a perfect square trinomial:
x2+12x+36=(x+6)2 x^2 + 12x + 36 = (x + 6)^2

STEP 4

Factor the second polynomial x2+9x+18x^2 + 9x + 18.
Look for factors of 18 that add up to 9:
x2+9x+18=(x+3)(x+6) x^2 + 9x + 18 = (x + 3)(x + 6)

STEP 5

Identify the unique factors from both factored polynomials:
From (x+6)2 (x + 6)^2 , the factor is x+6 x + 6 . From (x+3)(x+6) (x + 3)(x + 6) , the factors are x+3 x + 3 and x+6 x + 6 .

STEP 6

Determine the LCM by taking the highest power of each factor:
The LCM will include (x+6)2 (x + 6)^2 because it is the highest power of x+6 x + 6 from both polynomials. Also include (x+3) (x + 3) as it appears in the second polynomial.
Thus, the LCM is:
(x+6)2(x+3) (x + 6)^2(x + 3)

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