Problem 37 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{a^{3}-b^{3}}{a-b}$$

Step-by-Step Solution

Answer

The simplified form is \(a^{2} + ab + b^{2}\).

Step 1 - Understand the Problem

Given the rational expression \(\frac{a^{3}-b^{3}}{a-b}\), the goal is to reduce it to its lowest terms. Start by recognizing that the numerator is a difference of cubes.

Step 2 - Apply the Difference of Cubes Formula

Recall that the difference of cubes can be factored using the formula \(a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})\). Replace the numerator with its factored form so the expression becomes \(\frac{(a - b)(a^{2} + ab + b^{2})}{a - b}\).

Step 3 - Simplify the Expression

Notice that \(a - b\) appears in both the numerator and the denominator. As long as \(a eq b\), these factors can cancel out, leaving the expression \(a^{2} + ab + b^{2}\).

Step 4 - Present the Simplified Expression

After cancelling the common factors, the simplified form of the rational expression is \(a^{2} + ab + b^{2}\).

Key Concepts

Difference of Cubes

The difference of cubes is a special algebraic expression. It involves subtracting one cubed term from another. In simple terms, it's like taking away a three-dimensional block from another. The general form is:

\[a^3 - b^3 \]

The good news is that there's a standard formula to factor these types of expressions: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

Whenever you come across a difference of cubes, you can directly apply this formula. It breaks the expression into a simpler form that's easier to work with. Notice how it always results in one binomial \[ (a - b) \] and one trinomial \[(a^2 + ab + b^2) \].

Factoring Algebraic Expressions

Factoring algebraic expressions means breaking down a complex expression into simpler parts (factors). Think of it like splitting a piece of bread into manageable slices.

To factor the given expression \(\frac{a^3 - b^3}{a - b} \), identify the type of algebraic expression. Here, the numerator \(a^3 - b^3 \) is a difference of cubes. Using the formula from the previous section, we can factor it:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

So the rational expression becomes:
\[\frac{(a - b)(a^2 + ab + b^2)}{a - b} \]

Notice how \((a - b) \) appears in both the numerator and the denominator. They can cancel each other out, simplifying the expression.

Simplifying Fractions in Algebra

Simplifying fractions in algebra is similar to simplifying regular fractions. When you have a term in both the numerator and denominator, you can cancel them out.

In our example: \[\frac{(a - b)(a^2 + ab + b^2)}{a - b} \]

The factor \((a - b)\) cancels out because it is present in both parts of the fraction, as long as \(a eq b\). What remains is:
\[a^2 + ab + b^2 \]

This is the simplest form of the original rational expression. Learning to identify common terms and cancel them is a key part of simplifying fractions in algebra.

Always remember to check that the values you cancel do not make the original expression undefined. In this case, cancelling \(a - b \) is valid as long as \(a eq b \).