Problem 68 - Elementary Algebra

Question

For the following problems, reduce each rational expression if possible. If not possible, state the answer in lowest terms. \(\frac{x^{4}}{x^{5}-3 x}\)

Step-by-Step Solution

Answer

Question: Simplify the rational expression \(\frac{x^4}{x^5-3x}\). Answer: \(\frac{x^3}{x^4 - 3}\)

Step 1: Factor out the common factors from the numerator and denominator

Numerator: \(x^4\) The factors of \(x^4\) are \(x^4, x^3 \cdot x, x^2 \cdot x^2, x^2 \cdot x \cdot x, x \cdot x \cdot x \cdot x\) Denominator: \(x^5-3x\) We can factor out \(x\) from the denominator, so we get: \(x(x^4 - 3)\)

Step 2: Cancel out common factors, if any, and simplify the expression

We have the expression, \(\frac{x^4}{x(x^4 - 3)}\). We can see that \(x\) is a common factor in the numerator and the denominator. Cancel out the common factor \(x\): \((\frac{x^4}{x}) \div (\frac{x(x^4 - 3)}{x}) = \frac{x^3}{x^4 - 3}\) So our simplified expression is: \(\boxed{\frac{x^3}{x^4 - 3}}\) Since there are no more common factors between the numerator and the denominator, this is the final simplified expression, and the exercise is complete.

Key Concepts

Factoring

Factoring is a key skill in simplifying mathematical expressions, particularly when dealing with polynomials. Essentially, factoring means breaking down a complex expression into simpler components called "factors" that, when multiplied together, give back the original expression.
Think of it like breaking down a number into its prime factors. For example, if you have the number 12, you can factor it into \(2 \times 2 \times 3\).
When working with algebraic expressions like the numerator \(x^4\) or the denominator \(x^5 - 3x\), factoring allows us to identify parts that can be simplified. In our exercise, we see different ways to represent \(x^4\) through its factors:

• \(x^4\)
• \(x^3 \cdot x\)

This helps determine possible "cancellations" in the next steps of simplifying an expression.
Additionally, we factor the denominator \(x^5 - 3x\) by extracting the common factor \(x\), transforming the expression into \(x(x^4 - 3)\). Thus, factoring clears the path for further operations that simplify the expression considerably.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing the complexity of the fraction while ensuring that the value it represents does not change. This process often begins with factoring both the numerator and the denominator to identify any common factors.
For example, in the exercise \(\frac{x^4}{x^5 - 3x}\), the first step is factoring, as discussed before.

• The numerator is already factored as \(x^4\).
• The denominator can be factored to \(x(x^4 - 3)\).

Next, we identify and "cancel out" the common factors present in both the numerator and the denominator. In this case, \(x\) was common and thus canceled:
The simplification process involves reducing the expression to its lowest terms. This step ensures so that no more common factors remain. After simplifying \(\frac{x^4}{x(x^4 - 3)}\) by canceling the \(x\), it becomes \(\frac{x^3}{x^4 - 3}\). This is the simplest form. Simplifying algebraic fractions like this helps in solving equations and understanding relationships between variables more clearly.

Common Factors

Common factors are elements that are shared by two or more numbers or algebraic expressions. Discovering these can immensely help in the simplification process. For instance, in our exercise, both the numerator \(x^4\) and denominator \(x(x^4 - 3)\) shared the factor \(x\). This shared element is what we call a "common factor."
Recognizing common factors allows you to "cancel" them, streamlining the expression. It’s similar to simplifying numeric fractions. When you have \(\frac{8}{12}\), both 8 and 12 share 4 as a factor. Dividing both by 4, you get \(\frac{2}{3}\), which is a simpler form.
In terms of algebraic expressions:

• Identify the factors of each term, particularly in the numerator and denominator.
• Look for identical factors ("common factors").
• Divide both the numerator and the denominator by these common factors.

Once all common factors are canceled, you’re left with the simplest form of the expression, ensuring that no further reduction is possible. Thus, understanding and applying the concept of common factors effectively reduces algebraic fractions to their simplest form, making calculations easier and less error-prone.