Finding the least common multiple (LCM) of polynomials involves identifying the highest degree of each variable present in the polynomials and the highest coefficients. Let’s break it down step by step for the given polynomials: 9x2, 2x, and 3x2.
Step 1: Identify the Polynomials
The polynomials we are working with are:
- 9x2
- 2x
- 3x2
Step 2: Determine the Coefficients
The coefficients of the polynomials are:
- 9 for the polynomial 9x2
- 2 for the polynomial 2x
- 3 for the polynomial 3x2
Step 3: Find the LCM of the Coefficients
To find the LCM of the coefficients (9, 2, 3), we can list the multiples of each:
- Multiples of 9: 9, 18, 27, 36, …
- Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, …
- Multiples of 3: 3, 6, 9, 12, 15, 18, …
The least common multiple of 9, 2, and 3 is 18.
Step 4: Determine the Variables
Next, we evaluate the variables in each polynomial:
- In 9x2, the variable x has the highest exponent of 2.
- In 2x, the variable x has the exponent of 1.
- In 3x2, the variable x again has the exponent of 2.
The highest exponent of x among the three polynomials is 2.
Step 5: Compile the LCM of the Polynomials
The least common multiple of the polynomials can now be constructed using both the LCM of the coefficients and the highest exponent of variables:
LCM = 18x2
Conclusion
Therefore, the least common multiple of the polynomials 9x2, 2x, and 3x2 is:
18x2
This result allows you to efficiently work with expressions that contain these polynomials without worrying about their individual coefficients or variable powers!