Problem 37 - Algebra 2

Question

Find the LCM of each set of polynomials. $$ 9 p^{2} q^{3}, 6 p q^{4}, 4 p^{3} $$

Step-by-Step Solution

Answer

The LCM is \(36p^3q^4\).

Step 1: Identify the Highest Power of Each Variable in the Polynomials

For each variable that appears in the polynomials, identify the highest power of that variable.- For the variable \( p \), the highest power appearing is \( p^3 \) from \( 4p^3 \).- For the variable \( q \), the highest power appearing is \( q^4 \) from \( 6pq^4 \).

Step 2: Identify the Coefficients in Each Polynomial

Identify the coefficients for each polynomial:- The coefficients are 9, 6, and 4 respectively for the polynomials \( 9p^2q^3 \), \( 6pq^4 \), and \( 4p^3 \).

Step 3: Find the Least Common Multiple (LCM) of the Coefficients

Break down each coefficient into its prime factors:- 9 can be factored as \( 3 \times 3 \).- 6 can be factored as \( 2 \times 3 \).- 4 can be factored as \( 2 \times 2 \).Then, for LCM, take the highest power of each prime number:- For 2, the highest power is \( 2^2 \) (from 4).- For 3, the highest power is \( 3^2 \) (from 9).The LCM of the coefficients is \( 2^2 \times 3^2 = 36 \).

Step 4: Combine the Results

Combine the results obtained from Steps 1 and 3:- Use the highest powers of variables from Step 1: \( p^3 \) and \( q^4 \).- Use the LCM of the coefficients from Step 3: 36.Thus, the LCM of the polynomials is \( 36p^3q^4 \).

Key Concepts

Polynomial Coefficients

When working with polynomials, coefficients play a crucial role in determining the polynomial's characteristics. In simple terms, a coefficient is the numerical factor of a term in a polynomial. For example, in the term \( 9p^2q^3 \), the number 9 is the coefficient.

Understanding coefficients is important because they help in calculating the least common multiple (LCM) of polynomials.

• Identifying coefficients: For the polynomials \( 9p^2q^3 \), \( 6pq^4 \), and \( 4p^3 \), the coefficients are 9, 6, and 4 respectively.


• Role in LCM: When finding the LCM, breaking down coefficients into their prime factors helps in finding a common multiple that is used in the LCM calculation.

Thus, understanding polynomial coefficients is fundamental for problems like finding the LCM.

Highest Power Variable

Variables within polynomials are not only important, but the power to which they are raised also holds significance. The term "highest power variable" refers to the variable factor raised to the greatest power across similar terms.

This concept is crucial when determining the LCM of a set of polynomials. A good rule of thumb is:

• Find the maximum exponent for each variable appearing in the polynomials.


• For instance, with the polynomials given, the variable \( p \) reaches its highest power of \( p^3 \) in \( 4p^3 \), and \( q \) reaches \( q^4 \) in \( 6pq^4 \).

These highest powers ensure that every term in each polynomial can be fully expressed without omission after determining the LCM.

Prime Factorization

Prime factorization is an essential tool in mathematics, especially when finding the least common multiple of numbers, including polynomial coefficients.

Prime factorization involves breaking down a number into its smallest prime components, which allows for straightforward calculation of LCM:

• For example, the coefficient 9 breaks down into primes as \( 3 \times 3 \).


• Similarly, 6 can be expressed as \( 2 \times 3 \) and 4 as \( 2 \times 2 \).

Once factored, identifying the maximum occurrence of each prime helps in constructing the LCM, which is the basis for combining with the highest power variables for polynomial LCMs.

Polynomial Multiplication

Polynomial multiplication is the process of combining two or more polynomials, resulting in a single polynomial that contains all terms from the components. This topic ties closely with LCM, as it helps verify the inclusion of all polynomial terms.

Consider these steps in context:

• Combine the highest power variables: Using the polynomial with the highest degree for each variable ensures completeness.


• Multiply with the LCM of coefficients: The final product includes this, ensuring one single polynomial encompassing all terms.

With polynomial multiplication, the understanding of concepts like coefficients and highest power variables enables construction and verification of the combined polynomial, as seen in the solution \( 36p^3q^4 \).

Polynomial Coefficients

When working with polynomials, coefficients play a crucial role in determining the polynomial's characteristics. In simple terms, a coefficient is the numerical factor of a term in a polynomial. For example, in the term \( 9p^2q^3 \), the number 9 is the coefficient.

Understanding coefficients is important because they help in calculating the least common multiple (LCM) of polynomials.

• Identifying coefficients: For the polynomials \( 9p^2q^3 \), \( 6pq^4 \), and \( 4p^3 \), the coefficients are 9, 6, and 4 respectively.


• Role in LCM: When finding the LCM, breaking down coefficients into their prime factors helps in finding a common multiple that is used in the LCM calculation.

Thus, understanding polynomial coefficients is fundamental for problems like finding the LCM.

Highest Power Variable

Variables within polynomials are not only important, but the power to which they are raised also holds significance. The term "highest power variable" refers to the variable factor raised to the greatest power across similar terms.

This concept is crucial when determining the LCM of a set of polynomials. A good rule of thumb is:

• Find the maximum exponent for each variable appearing in the polynomials.


• For instance, with the polynomials given, the variable \( p \) reaches its highest power of \( p^3 \) in \( 4p^3 \), and \( q \) reaches \( q^4 \) in \( 6pq^4 \).

These highest powers ensure that every term in each polynomial can be fully expressed without omission after determining the LCM.

Prime Factorization

Prime factorization is an essential tool in mathematics, especially when finding the least common multiple of numbers, including polynomial coefficients.

Prime factorization involves breaking down a number into its smallest prime components, which allows for straightforward calculation of LCM:

• For example, the coefficient 9 breaks down into primes as \( 3 \times 3 \).


• Similarly, 6 can be expressed as \( 2 \times 3 \) and 4 as \( 2 \times 2 \).

Once factored, identifying the maximum occurrence of each prime helps in constructing the LCM, which is the basis for combining with the highest power variables for polynomial LCMs.

Polynomial Multiplication

Polynomial multiplication is the process of combining two or more polynomials, resulting in a single polynomial that contains all terms from the components. This topic ties closely with LCM, as it helps verify the inclusion of all polynomial terms.

Consider these steps in context:

• Combine the highest power variables: Using the polynomial with the highest degree for each variable ensures completeness.


• Multiply with the LCM of coefficients: The final product includes this, ensuring one single polynomial encompassing all terms.

With polynomial multiplication, the understanding of concepts like coefficients and highest power variables enables construction and verification of the combined polynomial, as seen in the solution \( 36p^3q^4 \).