Least Common Multiple (LCM) of Polynomials

Least Common Multiple (LCM) of Polynomials

The Least Common Multiple of two or more algebraic expressions is the expression of lowest degree (or power) such that the expressions exactly divide it.

Consider the following simple expressions a³b²,  a²b³ .

For these expressions LCM = a³b³.

To find LCM by factorization method

(i) Each expression is first resolved into its factors.

(ii) The highest power of the factors will be the LCM.

(iii) If the expressions have numerical coefficients, find their LCM.

(iv) The product of the LCM of factors and coefficient is the required LCM.

Example 3.12

Find the LCM of the following

(i) 8x⁴y², 48x²y⁴

(ii) 5x -10, 5x² – 20

(iii) x ⁴  -1, x ² − 2x + 1

(iv) x ³ - 27, (x - 3)², x ² – 9

Solution

(i) 8x ⁴y², 48x ²y⁴

First let us find the LCM of the numerical coefficients.

That is,      LCM (8, 48)  = 2 × 2 × 2 ×6 = 48

Then find the LCM of the terms involving variables.

That is, LCM (x ⁴y², x ²y⁴ )  = x ⁴y⁴

Finally find the LCM of the given expression.

We condclude that the LCM of the given expression is the product of the LCM of the numerical coefficient and the LCM of the terms with variables.

Therefore, LCM (8x ⁴y², 48x ²y⁴ ) = 48x ⁴y⁴

(ii) (5x -10), (5x ² - 20)

5x -10 = 5(x − 2)

5x ² - 20 = 5(x ² − 4) = 5(x + 2)(x − 2)

Therefore, LCM [(5x − 10),(5x ² −20)] = 5(x + 2)(x −2)

(iii) (x ⁴  -1),  x ² − 2x + 1

x⁴ -1  = (x ² )² −1 = (x ²  + 1)(x ² −1) = (x ²  + 1)(x + 1)(x −1)

x ² − 2x + 1 = (x −1)²

Therefore, LCM [(x ⁴ − 1),(x ² −2x +1)] = (x ²  + 1)(x + 1)(x −1)²

(iv) x ³ - 27, (x - 3)², x ² – 9

x ³ - 27 = (x − 3)(x ²  + 3x + 9) ; (x − 3)²  = (x − 3)² ; (x ² − 9) = (x + 3)(x − 3)

Therefore, LCM [(x ³ - 27),(x - 3)²,(x ² - 9)] = (x − 3)² (x + 3)(x ²  + 3x + 9)