Problem 49 - Algebra 2

Question

Find the LCM of each set of polynomials. \(9 x^{3}, 5 x y^{2}, 15 x^{2} y^{3}\)

Step-by-Step Solution

Answer

The LCM of the given polynomials is \(45x^3y^3\).

Step 1: Identify the highest powers of variables

For the LCM, we need to consider each variable present in all the polynomials and take the highest power of each. - For variable \(x\): - Polynomial 1: \(x^3\) - Polynomial 3: \(x^2\) Thus, the highest power is \(x^3\).- For variable \(y\): - Polynomial 2: \(y^2\) - Polynomial 3: \(y^3\) Thus, the highest power is \(y^3\).

Step 2: Identify the LCM of coefficients

Now determine the least common multiple of the numerical coefficients:- The coefficients are 9, 5, and 15.The LCM of these coefficients can be found by:- 9 = \(3^2\)- 5 = \(5^1\)- 15 = \(3^1 \times 5^1\)The LCM will be \(3^2 \times 5^1 = 45\).

Step 3: Combine the results

Formulate the LCM by combining the highest powers of variables with the LCM of coefficients:- From Step 1, we get \(x^3\) and \(y^3\).- From Step 2, we get coefficient 45.Thus, the LCM of the polynomials is \(45x^3y^3\).

Key Concepts

Highest Power of Variables

When determining the least common multiple (LCM) of polynomials, the highest power of each variable present in the polynomials plays a crucial role. The LCM should include each variable raised to the highest power that appears among the given set of polynomials.

To identify the highest power of variables, follow these steps:

• Look at each polynomial individually and note the power of each variable.
• For each distinct variable, find the maximum exponent across all polynomials.

Take, for example, the polynomials: \(9x^{3}\), \(5xy^{2}\), and \(15x^{2}y^{3}\). Here, the variable \(x\) appears as \(x^3\) and \(x^2\). Thus, the highest power for \(x\) is \(x^3\).
Similarly, the variable \(y\) appears in \(5xy^{2}\) as \(y^2\), and in \(15x^{2}y^{3}\) as \(y^3\). Thus, the highest power for \(y\) is \(y^3\).

By selecting these highest powers, we ensure that any polynomial in the set divides the LCM without leaving a remainder.

LCM of Coefficients

Apart from the variables, the coefficients of the polynomials must also be considered. The Least Common Multiple (LCM) of the numerical coefficients ensures that any constant factor in the polynomials is covered.

To find the LCM of coefficients, follow these steps:

• Factorize each coefficient into its prime factors.
• Identify the highest power of each prime factor that appears in any of the factorizations.
• Multiply these highest powers to get the LCM of the coefficients.

For example, given the coefficients 9, 5, and 15:

• 9 can be broken down as \(3^2\).
• 5 remains \(5^1\) as it is a prime number.
• 15 is \(3^1 \times 5^1\).

The LCM of these coefficients takes the highest power of each prime present: \(3^2\) and \(5^1\), resulting in an LCM of 45.

By obtaining the LCM of the coefficients, you ensure that the resultant polynomial coefficient is a multiple of each given polynomial's coefficient.

Polynomial Multiplication

The final step in finding the LCM of polynomials is combining the identified elements into a single polynomial using multiplication. This involves forming a new polynomial that captures both the variable part and the coefficient part.

Here's how to do it:

• Take the highest powers of the variables from the first step.
• Take the LCM of the coefficients from the second step.
• Multiply these two results together.

Using our example:

• The highest power of variable \(x\) was \(x^3\).
• The highest power of variable \(y\) was \(y^3\).
• The LCM of the coefficients was 45.

Combine these to get the LCM of the polynomials: \(45x^3y^3\).

By structuring polynomial multiplication in this organized manner, the LCM created is comprehensive. It ensures divisibility for each of the original polynomials and keeps operations straightforward.