LCM OF POLYNOMIALS BY FACTORING

The least common multiple of two or more polynomials is the expression of lowest degree (or power) such that the polynomials are exactly divided by it. 

Find the least common multiple of the following polynomials.

Question 1 :

4x²y, 8x³y²

Solution :

Let us factor each of the polynomials.

The highest power for 2 is 2², x is x³ and y is y²

L.C.M  =  2³⋅ x³ ⋅ y²   =  8 x³ y²

Hence the answer is 8 x³ y².

Question 2 :

-9a³b², 12a²b²c

Solution :

Let us factor each of the polynomials

L.C.M  =  2² ⋅ 3 ⋅ a² ⋅ b²⋅ c

  =  12a² b²c

Question 3 :

16m, -12m²n², 8n²

Solution :

16m  =  2⁴ ⋅ m

-12m²n²  =  -2² ⋅ 3 m²n²

8n² =  2³ ⋅ n² 

L.C.M  =  2⁴ m²n²

  =  16m²n²

Question 4 :

 p² − 3p +2, p² - 4

Solution :

p² − 3p + 2  =  p² − p - 2p + 2

  =  p(p - 1) - 2(p - 1)

=  (p - 1)(p - 2)

p² - 4  =  p² - 2²

=  (p + 2) (p - 2)

L.C.M  =  (p+2)(p-2)(p-1)

Question 5 :

2x² - 5x -3, 4x² -36

Solution :

2x² - 5x -3  =  2x² - 6x + 1x -3

  =  2x (x - 3) + 1(x - 3)

  =  (2x + 1)(x - 3)

4x² - 36  =  4(x² - 9)

  =  4(x² - 3)²

  =  4(x - 3)(x + 3)

L.C.M  =  4(2x + 1)(x + 3)(x - 3)

Question 6 :

(2x² -3xy)², (4x -6y)³, 8x³ -27y³

Solution :

(2x² -3xy)²  =  [x(2x - 3y)]²

  =  x²(2x - 3y)²

(4x -6y)³  =  [2(2x -3y)]³

  =  8(2x -3y)³

 8x³ -27y³  =  2³x³ - 3³ y³

  =  (2x)³ - (3y)³

  =  (2x - 3y) ((2x)² - (2x)(3y) + (3y)²)

  =  (2x - 3y) (4x² - 6xy + 9y²)

L.C.M  =  8x²(2x -3y)³(4x² - 6xy + 9y²)

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