Problem 66 - Elementary Algebra

Question

For the following problems, reduce each rational expression if possible. If not possible, state the answer in lowest terms. \(\frac{2 a^{3}+5 a}{a}\)

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{2a^3+5a}{a}\). Answer: The simplified expression is \(2a^2+5\).

Step 1: Factor the numerator and denominator

We start by factoring the numerator (if possible), and recognizing the factors in the denominator. In this case, the numerator is \(2a^3 + 5a\), and the denominator is \(a\). Factoring the numerator, we have: \(2a^3 + 5a = a(2a^2+5)\) The denominator is already in factored form.

Step 2: Cancel out the common factors

Now, we cancel the common factors in the numerator and denominator. The only common factor is \(a\): \(\frac{a(2a^2+5)}{a}=\frac{2a^2+5}{1}\)

Step 3: Simplify and write the answer in lowest terms

Since there are no more common factors to cancel, the expression is simplified and in lowest terms: \(\frac{2a^3+5a}{a} = 2a^2+5\)

Key Concepts

Factoring Polynomials

Breaking down a polynomial into simpler pieces, or factors, is a fundamental skill in algebra called factoring polynomials. Imagine you have a large box that's too bulky to transport. You can make it easier to carry by unpacking it into smaller boxes. Similarly, by factoring, you unveil the 'smaller boxes' or factors that multiply to give the original polynomial. This process is crucial when simplifying rational expressions, as you need to identify and cancel common factors.

Here’s how you might factor a simple polynomial: if you have a term like \(2a^3 + 5a\), it can be factored by extracting the common term, in this case, \(a\), resulting in \(a(2a^2 + 5)\). Recognizing these common terms is like finding the handle of each box, giving you something to hold onto when you move to the next step of simplification.

Canceling Common Factors

Once you have factored the polynomials, it's a bit like setting up a division problem where you can cancel out similar terms. This step is known as canceling common factors. In the context of our unpacking analogy, if you realize you’re carrying two boxes of the same weight and size to the same location, why not just carry one?

Cancellation in Action

In the given expression \(\frac{2a^3+5a}{a}\), after factoring the numerator, we can cancel the \(a\) in the numerator with the \(a\) in the denominator. This is because any number or term divided by itself is equal to one. Thus, \(\frac{a(2a^2+5)}{a}\) simplifies to \(\frac{2a^2+5}{1}\), or just \(2a^2+5\). This leaves us with a simpler expression, free of the original common factor.

Lowest Terms

The concept of expressing a fraction in lowest terms is similar to reducing the fraction to its simplest form; you want no excess baggage. Expressing rational expressions in lowest terms ensures there's no redundancy in the terms, making the expressions cleaner and easier to understand.

When you fully factor and cancel out all common factors between the numerator and denominator, like after canceling the \(a\) from both top and bottom of \(\frac{a(2a^2+5)}{a}\), you're left with \(2a^2+5\), which is in its lowest terms. No further simplification is possible, and it's as compact as it can get. Ensuring an expression is in its lowest terms can be a significant step in problem-solving as it makes the expression less complex and more manageable.

Algebraic Expressions

At the heart of simplifying rational expressions lie algebraic expressions, the phrases of the mathematical language that include numbers, variables (like \(a\)), and arithmetic operations. Understanding how to manipulate these expressions is critical for navigating through algebra.

An expression like \(2a^3+5a\) is an algebraic expression, as it combines variables and coefficients. Having a strong grasp on how these expressions operate allows students to progress to higher levels of math, where these foundational skills are applied to more complex concepts. By learning how to simplify algebraic expressions down to their simplest form, students can tackle more challenging problems with greater ease and clarity.