Question

Find the least common multiple of the expressions. \(24 x^2, 8 x^2-16 x\)

Short answer

The least common multiple of the expressions \(24 x^2\) and \(8 x^2-16 x\) is \(8x(x-2)\).

Step-by-step solution

Prime Factorization

First factorize the given expressions: \(24 x^2=2 x^2 * 2^3\) and \(8 x^2-16 x = 8x(x-2) = 2^3x * (x-2)\

Finding the LCM

Next, take the least power common to both expressions for each factor, yielding \(2^3x\). Then, take the highest power of the remaining factor which is \(x-2\).

LCM Expression

Finally, we combine these factors to form the least common multiple, \(2^3x(x-2)\) which simplifies to \(8x(x-2)\).

Key concepts

Polynomial Expressions

A polynomial expression is a mathematical phrase that can have variables, coefficients, and exponents. It combines them using addition, subtraction, multiplication, and non-negative integer exponents. By examining two polynomial expressions, such as those given in the exercise, we can understand how they are structured and how they behave. For example, in the expression \(24x^2\), the term \(24\) is the coefficient, \(x^2\) is the variable raised to a power, and the expression as a whole is a monomial because it consists of one term.
Another expression could be a binomial, meaning it consists of two terms added or subtracted, such as \(8x^2 - 16x\). To succinctly work with these expressions, one might use operations like factoring, which is crucial for determining the least common multiple. Simplifying polynomials often involves collecting like terms and factoring out common expressions to make operations easier.

Prime Factorization

Prime factorization is a method of breaking down a number or expression into its simplest building blocks – prime numbers or prime polynomial factors. It's like finding the unique recipe of prime numbers that multiply to get the original number. For the numbers in our exercise, like \(24\), we break it down into \(2^3 \times 3\), because these are the prime numbers that multiply to form \(24\).
In polynomial expressions, factorization might also involve finding common variables or expressions. For example, in \(8x^2 - 16x\), we can factor out \(8x\) as a common element, resulting in \(8x(x-2)\). The approach is similar to numeric factorization but involves variables too. This process simplifies expressions and allows us to identify common factors essential for finding the least common multiple.

Algebraic Expressions

Algebraic expressions, like polynomials, consist of integers, variables, and mathematical operations. They form the building blocks of algebra and can represent lots of different quantities. Handling algebraic expressions involves understanding how to manipulate them in forms that make calculations easier.
In real life, when you see expressions like \(24x^2\) or \(8x^2 - 16x\), you're dealing with components of larger algebraic equations. Simplifying them through operations like addition, subtraction, and factorization is crucial. For instance, to find the least common multiple of algebraic expressions, you want to see the biggest possible expression that each original expression can divide into evenly. Working with algebraic expressions was key in our exercise, where factorization and simplification led us to the common denominator or \(8x(x-2)\), representing the least common multiple.