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To find the least common multiple (LCM) of two or more polynomials, the polynomials must be factored completely. The LCM is the product of the factors with the highest power that appear in any of the polynomials. To show an example, the LCM of the following polynomials will be found. Polynomial I: & 12x^2y+48xy+48y Polynomial II: & 3x^3y-18x^2y-48xy The procedure of finding the LCM of the polynomials involves three steps.
Split into factors
Factor out 12y
a^2+2ab+b^2=(a+b)^2
Then the numerical factor can be written as a product of 2^2 and 3. Polynomial I [0.5em] 12x^2y+48xy+48y ⇕ 2^2 * 3 * y (x+2)^2 Now the other polynomial will be factored. Since the factor 3xy can be seen in each of its terms, start by factoring it out.
Split into factors
Factor out 3xy
| Standard Form | Factored Form | All Related Factors | |
|---|---|---|---|
| Polynomail I | 12x^2y+48xy+48y | 2^2 * 3 * y * (x+2)^2 | 2^2 * 3^1 * x^0 * y^() 1 * (x+2)^2 * (x-8)^1 |
| Polynomail II | 3x^3y-18x^2y-48xy | 3 * x * y * (x+2) * (x-8) | 2^0 * 3^1 * x^1 * y^() 1 * (x+2)^1 * (x-8)^1 |
The highest power of each prime factor can be listed as follows. 2^2, 3^1, x^1, y^1, (x+2)^2, and (x-8)^1