Problem 46 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{-x^{2}-2 x}{x^{2}-4}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-x}{x-2} \).

Step 1: Identify the expression

The expression to simplify is \( \frac{-x^{2}-2x}{x^{2}-4} \). Our goal is to simplify this fraction by factoring the numerator and the denominator.

Step 2: Factor the denominator

Observe that the denominator is \( x^{2} - 4 \), which is a difference of squares. This can be factored as: \[ x^2 - 4 = (x + 2)(x - 2) \]

Step 3: Factor the numerator

The numerator is \( -x^{2} - 2x \). We can factor this by taking \(-x\) as the common factor: \[ -x^{2} - 2x = -x(x + 2) \]

Step 4: Simplify the expression

Insert the factored forms into the original expression: \[ \frac{-x(x+2)}{(x+2)(x-2)} \] Since \(x + 2\) appears in both the numerator and the denominator, we can cancel it out, assuming \(x eq -2\):\[ \frac{-x}{x-2} \]

Step 5: Simplified expression

The simplified expression is \( \frac{-x}{x-2} \). Remember that the original expression is undefined when \(x = \pm 2\), but in the simplified expression, further undefined condition is only at \(x = 2\).

Key Concepts

Factoring

Factoring is an essential technique in algebra used to simplify expressions and solve equations.
To factor means to break down an expression into a product of its factors, which are simpler components that multiply together to give the original expression.

In the given problem, the numerator \(-x^{2} - 2x\) was factored by identifying the common factor \(-x\).
When you factor by grouping, you look for a term that is common across the terms in the expression.
For this problem, \(-x\) was the common factor in both \(-x^2\) and \(-2x\), allowing us to rewrite the expression as:

• \(-x(x + 2)\)

The denominator, \(x^2 - 4\), is easier to factor as it is a difference of squares, which we'll explore in detail in the next section. Recognizing these patterns is crucial to simplifying complex algebraic fractions effectively.

Difference of Squares

A difference of squares is a very important pattern in algebra. It refers to expressions of the form \(a^2 - b^2\).
These can always be rewritten as \((a + b)(a - b)\).
This pattern occurs frequently and can simplify many algebraic expressions.

In our problem, we identified \(x^2 - 4\) as a difference of squares.
This expression fits the form with \(a = x\) and \(b = 2\):

• \(x^2 - 4 = (x + 2)(x - 2)\)

Understanding this pattern allows for the quick and accurate factoring of specific quadratic expressions.
By memorizing the difference of squares, you can quickly factor expressions like these to simplify equations further.

Simplification Process

Simplifying algebraic fractions involves several steps, focusing on transforming an expression into its simplest form.
After factoring both the numerator and the denominator, the simplification process involves canceling out any common terms.

In the exercise's example, once the expression was rewritten with its factored terms as \(\frac{-x(x + 2)}{(x + 2)(x - 2)}\), we noticed that \(x + 2\) appeared in both the numerator and denominator.
This common factor can be canceled out; however, always consider the domains and restrictions of original variables.
When simplifying expressions, it’s crucial to note that factors can only be canceled if they are multiplied, not added or subtracted, within the expressions.

After canceling \(x + 2\), the simplified expression was obtained:

• \(\frac{-x}{x-2}\)

Hence, the fraction is now in its simplest form, assuming \(x eq 2\), since this value makes the original denominator zero. Always check for values that might unsettle the expression, especially when cancellation obscures potential restrictions.