Problem 45 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

\(\frac{3-x}{x}\)

Step 1: Factor the Numerator

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

Step 2: Factor the Denominator

The denominator is \(-x^2 - 3x\). Factor out \(-1\) first to make it easier to work with: \(-x^2 - 3x = -1(x^2 + 3x)\). Now, factor \(x\) out of \(x^2 + 3x\), which gives \(-1 \cdot x(x + 3)\). So the factored form is \(-x(x + 3)\).

Step 3: Simplify the Fraction

The simplified fraction is:\[\frac{(x-3)(x+3)}{-x(x + 3)}\]The \(x+3\) terms in the numerator and denominator cancel each other out, so:\[\frac{x-3}{-x}\]Simplify by factoring \(-1\) out of the denominator gives:\[-\frac{x-3}{x} = \frac{3-x}{x}\].

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a complex expression into simpler ones, usually as a product of its factors. Think of it as the process of reversing multiplication. In the example given, the polynomial \(x^2 - 9\) in the numerator needs factoring. Recognizing this as a "difference of squares" is the key.

• When you see a difference of squares, it takes the form \( a^2 - b^2 \), which can be rewritten as \( (a-b)(a+b) \).
• This means, for our example, \(x^2 - 9\) can be shown as \((x-3)(x+3)\), because 9 is \(3^2\).

Factoring turns expressions into simpler, separated terms, making simplifications like cancellations possible later on.

Difference of Squares

The difference of squares is a particular algebraic concept where you have two squared numbers subtracted from each other. It's a special case in polynomial factoring due to its structure, allowing it to be expressed as a product.

• The general pattern is \( a^2 - b^2 = (a-b)(a+b) \).
• Applying this to the example, \(x^2 - 9\) fits perfectly since it can be seen as \(x^2 - 3^2\).
• Thus, it factors into \((x-3)(x+3)\), breaking it into two binomials immediately.

By using this rule, algebraic simplification becomes far more manageable, specially when dealing with complex fractions or equations. Recognizing these patterns is a critical skill in algebra.

Simplifying Fractions

Simplifying fractions involves reducing the expression to its simplest form by cancelling out common factors from the numerator and the denominator. This provides clearer insight and often easier calculations. Here is how it's done:

• Begin with the factored expression, \(\frac{(x-3)(x+3)}{-x(x + 3)}\).
• Notice both the numerator and the denominator contain \(x+3\). You can cancel \(x+3\) from both parts.
• Once canceled, the expression becomes \(\frac{x-3}{-x}\).
• Simplify further by factoring out a \(-1\) from the denominator, which turns \(-x\) into \(-1\times x\). Thus transforming \(\frac{x-3}{-x}\) into \(-\frac{x-3}{x}\) or equivalently \(\frac{3-x}{x}\).

Reducing fractions helps streamline these algebraic expressions, cutting through complexity to reveal a straightforward form.