Problem 84 - Algebra and Trigonometry

Question

Simplify the expression. (This type of expression arises in calculus when using the “quotient rule.”) $$ \frac{2 x(x+6)^{4}-x^{2}(4)(x+6)^{3}}{(x+6)^{8}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{-2x(x-6)}{(x+6)^5}\).

Step 1: Understand the Problem

We are given the expression \( \frac{2x(x+6)^4 - x^2(4)(x+6)^3}{(x+6)^8} \). Our goal is to simplify this expression.

Step 2: Identify Common Factors

Observe that both terms in the numerator have \((x+6)^3\) as a common factor. We will factor \((x+6)^3\) out of the numerator.

Step 3: Factor Out the Common Factor

From the numerator \(2x(x+6)^4 - 4x^2(x+6)^3\), factor \((x+6)^3\). This results in \((x+6)^3 (2x(x+6) - 4x^2)\).

Step 4: Simplify Inside the Parentheses

Simplify the expression inside the parentheses: \(2x(x+6) - 4x^2\). Distribute to get \(2x^2 + 12x - 4x^2\). Combine like terms to get \(-2x^2 + 12x\).

Step 5: Rewrite the Expression

Substitute the simplified expression back into the factored form to get \(\frac{(x+6)^3 (-2x^2 + 12x)}{(x+6)^8}\).

Step 6: Simplify the Fraction

Since \((x+6)^3\) is in both the numerator and the denominator, they cancel out, leaving \(\frac{-2x^2 + 12x}{(x+6)^5}\).

Step 7: Factor the Numerator

Factor the remaining numerator \(-2x(x - 6)\), resulting in \(\frac{-2x(x-6)}{(x+6)^5}\).

Key Concepts

Quotient Rule in Calculus

The quotient rule is a technique used for differentiating functions in calculus. It's particularly useful when you have a fraction where both the numerator and the denominator are functions of a variable. The quotient rule formula is:

• If you have a function given by \( \frac{u(x)}{v(x)} \), the derivative \( \left( \frac{u}{v} \right)' \) is \( \frac{u'v - uv'}{v^2} \).
• Here, \( u(x) \) and \( v(x) \) are functions with respect to \( x \), while \( u' \) and \( v' \) are their respective derivatives.

In the context of simplifying rational expressions, the quotient rule conceptually parallels the simplification process. Simplifying involves canceling common factors in the numerator and denominator, assisting in smoothing out complex expressions. This aligns well with the overarching goal of calculus to simplify expressions during differentiation.
Remember, simplifying expressions is key before applying the quotient rule, as it prevents unnecessary complexity and errors.
This simplification, like in the original exercise, ensures calculations remain manageable and accurate.

Factoring Polynomial Expressions

Factoring is a pivotal skill in algebra, especially when simplifying polynomial expressions, similar to how the original exercise required the expression in the numerator to be factored.
Factoring involves rewriting a polynomial as a product of its simpler factors. This step often reveals common factors that can simplify the expression further.To factor polynomial expressions, consider:

• Looking for and identifying common factors in terms and expressions.
• Using special factoring formulas, like the difference of squares or perfect square trinomials.
• Applying strategic methods such as grouping terms or using the factor theorem for more complex expressions.

In our given exercise, factoring out \((x+6)^3\) from the terms in the numerator allowed us to greatly simplify the expression.
The simplified expression, \(2x(x+6) - 4x^2\), was further simplified to reveal deeper commonalities which were utilized in simplifying the entire rational expression.
Factoring is essential as it sometimes uncovers hidden patterns, making problems easier to solve and understand.

Common Factors in Algebra

Spotting common factors from algebraic expressions is a crucial step in simplifying equations or expressions. A common factor is an identical element present in two or more terms of an expression.

• Identifying these common elements can reduce complexity by canceling them out, just as seen in the steps of the original solution.
• When dealing with expressions, always look for shared components across terms to factor them out.

For instance, in the exercise, (x+6)^3 was a common factor in the numerator.
By factoring it out, we could cancel it against the denominator, thus simplifying the original cumbersome expression into something more manageable.
Keeping an eye out for such factors allows for streamlined calculations and aids in solving algebraic problems with greater ease and accuracy.
Mastering the art of spotting and factoring common components builds a solid foundation for tackling more complex algebraic and calculus problems.