Problem 3 - Elementary Algebra

Question

Perform the following divisions. $$ \frac{a^{2} b+3 a b^{2}+2 b}{a b} $$

Step-by-Step Solution

Answer

Question: Simplify the following algebraic expression: $$\frac{a^2 b + 3ab^2 + 2b}{ab}$$ Answer: The simplified expression is: $$a + 3 b + \frac{2}{a}$$

Step 1: Identify the Terms in the Expression

First, let's identify the terms in the given expression. There are three terms in the numerator: 1. \(a^{2} b\) 2. \(3 a b^{2}\) 3. \(2 b\) And one term in the denominator: 1. \(a b\) Our task is to divide each of the terms in the numerator by the term in the denominator.

Step 2: Divide the First Term

Now, let's divide the first term in the numerator (\(a^{2} b\)) by the denominator (\(a b\)). $$ \frac{a^{2} b}{a b} $$ The \(a\) in the numerator cancels out with the \(a\) in the denominator, and the \(b\) in the numerator cancels out with the \(b\) in the denominator, leaving us with: $$ a $$

Step 3: Divide the Second Term

Next, let's divide the second term in the numerator (\(3 a b^{2}\)) by the term in the denominator (\(a b\)). $$ \frac{3 a b^{2}}{a b} $$ Again, the \(a\) in the numerator cancels out with the \(a\) in the denominator. Additionally, one of the \(b\)'s in the numerator cancels out with the \(b\) in the denominator, leaving us with: $$ 3 b $$

Step 4: Divide the Third Term

Finally, let's divide the third term in the numerator (\(2 b\)) by the term in the denominator (\(a b\)). $$ \frac{2 b}{a b} $$ The \(b\) in the numerator cancels out with the \(b\) in the denominator, leaving us with: $$ \frac{2}{a} $$

Step 5: Combine the Results

Now that we've divided each term individually, let's combine the results to obtain the final simplified expression. We combine the results of step 2, step 3, and step 4 as follows: $$ a + 3 b + \frac{2}{a} $$ Thus, the simplified expression is: $$ a + 3 b + \frac{2}{a} $$

Key Concepts

Simplifying Expressions

Simplifying expressions is a key skill in algebra that helps make equations more manageable. It involves the process of rewriting an expression in its simplest form. This is done by combining like terms, canceling terms, or factoring when necessary.

In the context of the given problem, we're asked to simplify a fraction where the numerator is a polynomial and the denominator is a term. Simplifying such expressions often involves dividing each term in the numerator by the term in the denominator. This simplifies the expression from a complex polynomial form into a cleaner, more understandable format.

By ensuring that the terms are fully simplified, you make it easier to perform further calculations or solve equations that may use this expression later. Let's look at the other important aspects of the solution process.

Polynomial Division

Polynomial division works similarly to regular arithmetic division but involves dividing polynomials rather than numbers. It allows us to divide each term in the polynomial separately, which is crucial for breaking down complex expressions into simpler terms.

In our specific example, the polynomial in the numerator consists of three terms: \( a^{2} b, 3ab^{2}, \) and \( 2b \). The term in the denominator is \( ab \).

To divide the polynomial, we take each term from the numerator and divide it by the term in the denominator:\[\frac{a^{2} b}{a b}, \frac{3 a b^{2}}{a b}, \text{ and } \frac{2 b}{a b}\]

This step-by-step division allows us to simplify each term individually. Understanding this principle can often make polynomial division easier and less intimidating. Remember, the goal is always to rewrite the expression in a much simpler form.

Canceling Terms

Canceling terms is a method used during polynomial division to simplify expressions even further. This technique involves removing common factors from both the numerator and the denominator.

Take a look at the division of the first term, \( \frac{a^{2} b}{a b} \). We cancel out the \( a \) and \( b \) from both the numerator and the denominator, leaving us with \( a \). A similar approach can be observed with the other terms as well. For \( \frac{3 a b^{2}}{a b} \), the \( a \) from the numerator and the denominator cancels, and one \( b \) cancels out from both sides leaving \( 3b \). This practice results in significantly simplified terms for each fraction.

Understanding how to recognize and cancel common terms is crucial. It not only simplifies the given expression but also makes it easier to perform further mathematical operations. Remember, the effectiveness of canceling terms requires a thorough check for the same common factors present in both the numerator and the denominator.