Problem 24 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{9-x^{2}}{3 x^{3}-27 x}$$

Step-by-Step Solution

Answer

\(-\frac{1}{3x}\) is the simplified form.

Step 1: Factor the Numerator

The numerator is \(9 - x^2\), which can be written as a difference of squares. It can be factored as \((3 - x)(3 + x)\).

Step 2: Factor the Denominator

The denominator is \(3x^3 - 27x\). Start by factoring out the greatest common factor, which is \(3x\). So the expression becomes \(3x(x^2 - 9)\). Notice \(x^2 - 9\) is a difference of squares, which can be factored further into \((x - 3)(x + 3)\). This makes the full factorization \(3x(x - 3)(x + 3)\).

Step 3: Cancel Common Factors

Both the numerator and the denominator have the common factors \((3 + x)\) and \((3 - x) = -(x - 3)\). Cancel out these common factors. After cancellation, you're left with \(-1/(3x)\).

Step 4: Simplify the Result

The simplified expression after canceling the common factors is \(-\frac{1}{3x}\).

Key Concepts

Factoring

Factoring is like breaking down a number or expression into its simplest pieces (or factors) that multiply to give you the original number or expression. For algebraic expressions, this process helps us simplify the expressions greatly.
In the context of this exercise, we start by factoring both the numerator and the denominator of an algebraic fraction. Consider the numerator, which is the expression \(9 - x^2\). By recognizing it as a special type of expression called the 'difference of squares', we apply a known formula:

• The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\).

So, \(9 - x^2\) is factored into \((3 - x)(3 + x)\).
Likewise, for the denominator \(3x^3 - 27x\), begin by finding and factoring out the greatest common factor, which is \(3x\). This step brings the expression to \(3x(x^2 - 9)\). We can also apply the difference of squares to \(x^2 - 9\), which factors further to \((x - 3)(x + 3)\).
This process of factoring helps simplify complex expressions, making them easier to manipulate and understand!

Difference of Squares

The difference of squares is an important algebraic concept where you have a binomial expression like \(a^2 - b^2\), which can be factored into two products: \((a - b)(a + b)\). This particular technique is valuable because it transforms a subtracted squared term into a multiplication of linear terms, simplifying many problems.
In this exercise, we applied this concept twice:

• First, to the numerator \(9 - x^2\), transforming it into \((3 + x)(3 - x)\).
• And again to \(x^2 - 9\) in the denominator, factoring it into \((x - 3)(x + 3)\).

By using the difference of squares, large and complex polynomial expressions can often be broken down into their simpler factors. Recognizing when this formula applies is crucial for simplifying expressions in algebra.

Polynomial Division

When simplifying algebraic fractions, especially after factoring, polynomial division comes into play with the cancellation of common terms. Division in algebra isn't merely dividing numbers but involves strategically eliminating terms.
In this particular exercise, after factoring both the numerator \( (3 - x)(3 + x) \) and the denominator \( 3x(x - 3)(x + 3) \), you observe common terms between them.

• Specifically, the terms \(3 + x\) and \((3 - x) = -(x - 3)\) appear in both the numerator and denominator.

These common terms can be cancelled, assuming they are not equal to zero, as division by zero is undefined in mathematics.
What remains after this cancellation is a simpler form of the fraction, leading to the expression \(-\frac{1}{3x}\). This final expression is much simpler, showing how polynomial division and cancellation are key to simplifying algebraic fractions effectively.