Problem 48 - Algebra 2

Question

Find the LCM of each set of polynomials. \(15 a b^{2} c, 6 a^{3}, 4 b c^{2}\)

Step-by-Step Solution

Answer

The LCM of the polynomials is \(60 a^3 b^2 c^2\).

Step 1: Identify the Prime Factors of Each Coefficient

For the numbers 15, 6, and 4, break them down into prime factors:- 15 factors as \(3 \times 5\).- 6 factors as \(2 \times 3\).- 4 factors as \(2^2\).

Step 2: Identify the Highest Power for Each Prime Number

The LCM takes each prime number at its highest power:- Prime 2: The highest power is \(2^2\).- Prime 3: The highest power is \(3\).- Prime 5: The highest power is \(5\).

Step 3: Multiply the Highest Powers of the Primes

Calculate the product of the highest powers: \[LCM( ext{coefficients}) = 2^2 \times 3 \times 5 = 60\].

Step 4: Identify Each Variable's Highest Degree in the Polynomials

List the variables and their highest powers from each polynomial:- Variable \(a\): The highest power is \(a^3\).- Variable \(b\): The highest power is \(b^2\).- Variable \(c\): The highest power is \(c^2\).

Step 5: Combine the Coefficients and Variables

Form the LCM by multiplying the coefficient LCM and the highest powers of the variables:\[LCM = 60 \times a^3 \times b^2 \times c^2\].

Step 6: Conclusion

The least common multiple of the polynomials is \(60 a^3 b^2 c^2\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its simplest building blocks—prime numbers. Prime numbers are integers greater than 1 that have no divisors other than 1 and themselves. For example, the number 15 can be factored into prime numbers as \(3\times5\). Similarly, 6 can be factored as \(2\times3\) and 4 as \(2^2\).

Once we identify the prime factors of each coefficient in a polynomial, we look for the largest power of each prime factor. This is crucial for the calculation of the least common multiple (LCM), as the LCM uses each prime number raised to its highest power observed in any of the numbers. By identifying these highest powers, we ensure that our LCM is indeed the least common, or smallest value that all numbers can divide without a remainder.

Polynomials

Polynomials are algebraic expressions that consist of variables, coefficients, and exponents. They can have one or multiple terms, and each term is a product of a constant coefficient and one or more variables raised to an exponent.

When we find the LCM of a set of polynomials, we are looking for a polynomial that each original polynomial can divide without remainder. In our exercise involving polynomials \(15ab^2c\), \(6a^3\), and \(4bc^2\), each term needs to be broken into individual parts to discover its overall influence on the form of the LCM. This involves understanding how each variable and each coefficient contributes to the structure of the polynomial as a whole.

Highest Power

The term 'highest power' refers to the largest exponent that a prime number or variable appears with in the given numbers or polynomials. Calculating the LCM involves identifying these highest powers to ensure that the result is divisible by each term in the original set.

For instance, the prime factor 2 appears as \(2^2\) in some numbers, while it may only appear as \(2\) in others. The highest form, \(2^2\), is what we use when calculating the LCM. Similarly, for the variables in our polynomial exercise, the highest power of each variable across all the terms is extracted. For variable \(a\), this is \(a^3\), for \(b\) it's \(b^2\), and for \(c\), it's \(c^2\). By using the largest exponents from each of the terms, our calculation ensures full representation for each polynomial.

Variables

Variables are symbols used to represent numbers in algebraic expressions and are often associated with coefficients and exponents in polynomials. In the context of finding an LCM of polynomials, understanding how to handle variables is crucial.

Each variable in a polynomial can appear with different powers across terms. To determine the LCM, examine each variable separately and note its highest power found in any of the terms. This ensures that every polynomial divides the resulting LCM without remainder. In our specific example, the variable \(a\) goes up to \(a^3\), \(b\) to \(b^2\), and \(c\) to \(c^2\). These powers are then included in the LCM computation, making sure that the final result, such as \(60a^3b^2c^2\), encompasses all variables and powers involved in the original polynomials.