Problem 32 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{b^{8}-a b^{5}}{a b^{5}}$$

Step-by-Step Solution

Answer

\(\frac{b^3 - a}{a}\)

Step 1: Factor the Numerator

Identify common factors in the numerator. The numerator is \(b^8 - ab^5\). Notice that both terms share a common factor of \(b^5\). Factor \(b^5\) out from both terms, resulting in \(b^5 (b^3 - a)\).

Step 2: Rewrite the Expression

Rewrite the given rational expression using the factored numerator. This gives us \(\frac{b^5 (b^3 - a)}{ab^5}\).

Step 3: Cancel Common Factors

Cancel the common factor of \(b^5\) from the numerator and the denominator. We are left with \(\frac{b^3 - a}{a}\).

Key Concepts

Factoring Polynomials

Factoring polynomials is one of the first steps in simplifying complex algebraic expressions. Essentially, factoring involves breaking down a polynomial into simpler terms that, when multiplied together, give you the original polynomial. In the given problem, you have the polynomial expression in the numerator: \[ b^8 - ab^5 \].To factor this, look for common factors in each term. Here, both terms include a factor of \( b^5 \). Extract \( b^5 \) from the expression, resulting in \( b^5( b^3 - a )\). Breaking this down makes it much easier to handle and sets you up to simplify the rational expression further.

Canceling Common Factors

Once you've factored the numerator, the next step involves canceling out common factors in both the numerator and the denominator. After factoring the numerator to \( \frac{b^5(b^3 - a)}{ab^5} \), you can easily see that \( b^5 \) is present in both the numerator and the denominator. Because these factors are the same, they can be canceled out.Canceling common factors is similar to canceling out equal values in fractions. Just like how \( \frac{4}{8} \) simplifies to \( \frac{1}{2} \) by canceling out the common factor of 4, we can cancel \( b^5 \) in our expression to simplify it. This leaves us with \( \frac{b^3 - a}{a} \).

Simplifying Algebraic Fractions

After canceling common factors, your expression is now simpler. Simplifying algebraic fractions involves putting the expression into its simplest form by making sure no common factors are left in the numerator and the denominator.From the previous steps, we have reduced \( \frac{b^5(b^3 - a)}{ab^5} \) to \( \frac{b^3 - a}{a} \). At this point, check whether the numerator and the denominator share any further common factors that can be canceled out. In our example, they do not. Hence, \( \frac{b^3 - a}{a} \) is the simplest form of the original rational expression.This simplified form is easier to work with in further mathematical operations, allowing you to focus on solving problems rather than getting bogged down by complex terms.