Problem 86 - Precalculus: Functions and Graphs

Question

Simplify the expression. $$\frac{\left(x^{2}-5\right)^{4}\left(3 x^{2}\right)-x^{3}(4)\left(x^{2}-5\right)^{3}(2 x)}{\left[\left(x^{2}-5\right)^{4}\right]^{2}}$$

Step-by-Step Solution

Answer

\( \frac{-5x^4 - 15x^2}{(x^2 - 5)^5} \)

Step 1: Simplify the Numerator

Begin by simplifying the numerator, which is \( \left(x^2 - 5\right)^4 (3x^2) - x^3 (4) (x^2 - 5)^3 (2x) \). Factor out the common term \( (x^2 - 5)^3 \):\[= (x^2 - 5)^3 \left[ \left(x^2 - 5\right)(3x^2) - x^3 (8x) \right]\]Combine and simplify the terms inside the brackets:\[= (x^2 - 5)^3 \left[ 3x^4 - 15x^2 - 8x^4 \right]\]Combine like terms:\[= (x^2 - 5)^3 (-5x^4 - 15x^2)\]Factor out \(-5x^2\):\[= -5x^2(x^2 - 5)^3 (x^2 + 3)\]

Step 2: Simplify the Denominator

Now, observe the denominator: \(\left[\left(x^2 - 5\right)^4\right]^2\).This equates to:\[= \left(x^2 - 5\right)^{8}\]

Step 3: Combine and Simplify the Expression

With the simplified numerator and denominator, the expression becomes:\[\frac{-5x^2 (x^2 - 5)^3 (x^2 + 3)}{(x^2 - 5)^{8}}\]Simplify by reducing the common terms. Cancel \((x^2 - 5)^3\) in the numerator with \((x^2 - 5)^8\) in the denominator:\[= \frac{-5x^2 (x^2 + 3)}{(x^2 - 5)^{5}}\]

Step 4: Final Simplification

To further simplify, distribute \(-5x^2\) in the numerator:\[= \frac{-5x^4 - 15x^2}{(x^2 - 5)^5}\]

Key Concepts

Simplification

When we talk about simplifying rational expressions, the idea is to make the expression as concise as possible without changing its value. This often involves reducing fractions by identifying common factors in the numerator and the denominator.

• Identify Common Parts: Look for elements that occur in both the numerator and the denominator.
• Cancel Out: Reduce the expression by canceling matching factors in both parts.
• Simplify Inside Parentheses: If parentheses are present, first simplify the expression inside them.

Breaking down our example, we started with an expression that was quite lengthy. By recognizing the repeated term \(x^2 - 5\), we were able to simplify the numerator step-by-step. Remember, simplification often requires patience and careful observation to spot all common factors and terms that can be reduced.

Factoring

Factoring is indispensable when simplifying rational expressions as it allows us to transform expressions into products of simpler expressions. This can highlight potential common factors between the numerator and the denominator.

• Common Factor Identification: Identify terms that appear in every part of the expression, making them excellent candidates for factoring out.
• Application: In our given problem, notice how \((x^2 - 5)^3\) could be factored out from both terms in the numerator.
• Factor by Grouping: Sometimes grouping terms before factoring can simplify the process.

The core principle is to express the polynomial as a product using its factors. Once this is achieved, simplifying the entire expression becomes a much more manageable task.

Exponentiation

Exponentiation involves raising numbers or expressions to a power, which is a key element in our exercise. Understanding how to simplify different powers is crucial when working with expressions.

• Power of a Power Rule: In cases such as our expression, applying rules like \((a^m)^n = a^{m \times n}\) can significantly simplify calculations.
• Combine Like Terms: Powers of the same base can sometimes be combined or cancelled, as we do with the terms \(((x^2 - 5)^4)^2 = (x^2 - 5)^8\).
• Negative and Zero Exponents: These special cases can further reduce expressions under certain conditions.

Exponentiation provides a compact way to represent repeated multiplication and must be dealt with carefully to maintain the precision of our expressions throughout the problem-solving process.

Polynomial Division

Polynomial division, which appears in our exercise in the form of simplifying fractions with polynomials, involves dividing both the numerator and denominator by their common terms or factors.

• Simplify Through Cancellation: Division in rational expressions often reduces to cancelling common factors, as we did with \((x^2 - 5)^3\) in the numerator and \((x^2 - 5)^8\) in the denominator.
• Long Division vs. Synthetic Division: For more complex polynomials, methods like long division or synthetic division might be necessary.
• Always Reduce: After every division, try to simplify the result as much as possible by finding other common factors.

This process not only simplifies but also standardizes expressions, presenting them in their simplest form. It streamlines further evaluation and understanding of the expression, making it easier to work with in subsequent steps or applications.