EDGE Math 8

The LCM of two polynomials is obtained using the same procedure as getting the LCM of two numbers.

Example

Find the LCM of each set of polynomials.

1. $12 x and 15 y$

2. $3 m, 12 m^{2}, and 4 m^{3}$

3. $x^{2} - 4 and x^{2} - 4 x + 4$

Solution

1. First, write the prime factorization of each.

$12 x = 2 \cdot 2 \cdot 3 \cdot x$

$15 y = 3 \cdot 5 \cdot y$

The GCF of $12 x$ and $15 y$ is 3. Thus, their LCM is

$3 \cdot 2 \cdot 2 \cdot 5 \cdot x \cdot y = 60 x y .$

2. First, write the prime factorization of each.

$3 m = 3 \cdot m$

$12 m^{2} = 2 \cdot 2 \cdot 3 \cdot m \cdot m$

$4 m^{3} = 2 \cdot 2 \cdot m \cdot m \cdot m$

The GCF of $3 m,$ $12 m^{2},$ and $4 m^{3}$ is m. Thus, their LCM is

$m \cdot 2 \cdot 2 \cdot 3 \cdot m \cdot m = 12 m^{3} .$

3. First, write the prime factorization of each.

$x^{2} - 4 = (x + 2) (x - 2)$

$x^{2} - 4 x + 4 = {(x - 2)}^{2}$

The GCF of the two polynomials is $(x - 2) .$ Thus, their LCM is

(x - 2) (x + 2) (x - 2)

= (x + 2) {(x - 2)}^{2} .