Problem 21 - Introductory Algebra for College Students

Question

$$\frac{x^{2}+5 x+6}{x^{2}+x-6} \cdot \frac{x^{2}-9}{x^{2}-x-6}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{(x+3)}{(x-2)} \)

Step 1: Factorise Quadratic Expressions

Factorise each of the quadratic expressions, we have: \( \frac{(x+2)(x+3)}{(x-2)(x+3)} \cdot \frac{(x+3)(x-3)}{(x-3)(x+2)} \)

Step 2: Cancel Out Common Terms

Simplify the expressions by cancelling out the common terms, which gives \( \frac{(x+2)}{(x-2)} \cdot \frac{(x+3)}{(x+2)} \)

Step 3: Further Simplification

Simplify the expression further by cancelling out (x+2) in the numerator and denominator, this gives \( \frac{(x+3)}{(x-2)} \)

Key Concepts

Simplifying Algebraic Fractions

When simplifying algebraic fractions, our goal is to reduce the expression to its simplest form, much like you would simplify a numerical fraction like \(\frac{12}{16}\) to \(\frac{3}{4}\) by dividing both the numerator and denominator by their greatest common divisor, which in this case is 4. Similarly, in algebra, we look for common factors in the polynomials that make up the numerator and the denominator.

Simplifying begins with factoring these polynomials to reveal if they have any factors in common. In the original exercise, the factoring step exposes these common factors. Once you've factored the expression, the next crucial step is to identify and cancel out terms that appear in both the numerator and the denominator because any term divided by itself equals one. For example, with the fraction \(\frac{ab}{ac}\), you can cancel out the 'a' to simplify it to \(\frac{b}{c}\). This rule of canceling common factors applies to both numeric and algebraic fractions.

Canceling Common Terms in Algebra

Canceling common terms in algebra follows the principle of simplifying expressions by eliminating factors that appear in both the numerator and the denominator. It is important to note that terms can only be canceled if they appear as factors, meaning they are being multiplied together, not when being added or subtracted.

In our example, once the quadratic expressions are factored, you can clearly see pairs of common factors such as \((x+2)\) and \((x+3)\). Because these factors are the same in both the numerator and the denominator, they 'cancel out' to 1. After canceling the common factors, the expression often looks much simpler. Canceling is not just a cosmetic change; it fundamentally simplifies expressions, and is an essential step for solving algebraic equations or inequalities efficiently. One must always be careful to cancel only common factors and not terms out of context, which is a common mistake.

Quadratic Expressions Step-by-Step Solution

A step-by-step solution to a problem involving quadratic expressions usually includes several stages, starting with factoring, then simplifying by canceling, and finally, computing the result if necessary. Factoring quadratic expressions is about rewriting the quadratic polynomial as a product of its factors.

Let's use our textbook problem as an illustration. We begin by factoring the quadratic expressions in both the numerator and the denominator. This step requires knowledge of patterns, such as difference of squares shown by \((x^2 - 9)\to(x+3)(x-3)\), or trinomial factoring, for example \((x^2 + 5x + 6)\) factoring to \((x+2)(x+3)\).

After factoring, we proceed to simplify the expression by canceling out terms that are common to both the numerator and the denominator. In the final stage, the expression is streamlined by removing these common factors to reveal the simplified form which, for our exercise, resulted in \(\frac{(x+3)}{(x-2)}\). This process provides a clear, step-by-step path from a complex expression to an elegant, simplified result.