Question

Find the least common multiple of the expressions. \(x^2+3 x-40, x-8\)

Short answer

The least common multiple of the expressions \(x^2+3x-40\) and \(x-8\) is \((x-5)(x+8)(x-8)\).

Step-by-step solution

Simplify the First Expression

We should first simplify the expression \(x^2+3x-40\) by factoring it. The factors of -40 that add up to 3 are 8 and -5. Therefore, we can write the first expression as \((x-5)(x+8)\).

Analysis of Result

We can observe that the factor \(x-8\) from the second polynomial is not present in the factored form of the first expression. Therefore, the first expression is not divisible by the second.

Least Common Multiple

Since \(x^2+3x-40\) is not divisible by \(x-8\), their least common multiple (LCM) is simply the product of both. Hence, \(LCM(x^2+3x-40, x-8) = (x^2+3x-40)(x-8)=(x-5)(x+8)(x-8)\).

Key concepts

Factoring Polynomials

Factoring polynomials can feel like solving a puzzle. It's all about breaking down a complex expression into simpler components.
To start, ask yourself: Can this polynomial be expressed as a product of simpler polynomials? If yes, you've got yourself a factorable polynomial. In our example, we were given the quadratic expression \(x^2 + 3x - 40\). When factoring, the goal is to find two numbers that not only multiply to \(-40\) but also add up to the middle coefficient, which is \(3\) here.
This leads us to the numbers \(8\) and \(-5\).The expression \(x^2 + 3x - 40\) can then be written as \((x+8)(x-5)\). By writing it this way, we've simplified the expression into its factors.
This makes it easier to find the least common multiple and to solve other mathematical problems.

Polynomial Expressions

Understanding polynomial expressions is key to delving into algebra. Polynomials are expressions made up of variables and constants combined using addition, subtraction, multiplication, and non-negative integer exponents.
They can look pretty much like any combination of terms; for example, a simple polynomial expression is \(x^2 + 3x - 40\).Each part of a polynomial, like \(x^2, 3x,\) or \(-40\), is called a term. Notice how each term can vary in complexity, from just a constant (\(-40\)) to a product of a coefficient and a variable raised to an exponent (\(x^2\)).Expressing polynomials in factored form helps us with operations like finding the LCM.
It's because factoring reveals the core building blocks of an expression, much like uncovering the vocabulary of a language.

Mathematics Problem Solving

Mathematics problem solving is often about breaking a problem into manageable parts. Here, in the problem about Least Common Multiple (LCM), starting with what's given is crucial. We have two expressions to manage: \(x^2 + 3x - 40\) and \(x - 8\).The first step is to factor where possible. Doing this simplifies expressions and makes it clear what the common factors are. We found that \(x^2 + 3x - 40\) factors into \((x-5)(x+8)\). Once we have these factors, we look for overlapping components between this and the linear polynomial \(x - 8\). If there are none, like in this exercise, find the LCM by multiplying the distinct factors together.Thus, understanding each step and knowing how to piece the puzzle together culminates in solving what's been asked, leading to an answer that's both accurate and simplified.