Problem 28 - College Algebra

Question

Multiply or divide as indicated. $$ \frac{x^{2}+x}{x^{2}-4} \div \frac{x^{2}-1}{x^{2}+5 x+6} $$

Step-by-Step Solution

Answer

Therefore, \( \frac{{x^{2} + x}}{{x^{2} - 4}} \div \frac{{x^{2} - 1}}{{x^{2} + 5x + 6}} = \frac{{x(x + 3)}}{{(x - 2)(x - 1)}} \)

Step 1 Factorize the Expressions

Start by factorizing every quadratic expression in the fractions: \( \frac{{x(x + 1)}}{{(x - 2)(x + 2)}} \div \frac{{(x - 1)(x + 1)}}{{(x + 2)(x + 3)}} \)

Step 2 Convert Division to Multiplication

Remember that dividing by a fraction is the same as multiplying by its reciprocal. Hence, the expression will be converted to: \( \frac{{x(x + 1)}}{{(x - 2)(x + 2)}} \cdot \frac{{(x + 2)(x + 3)}}{{(x - 1)(x + 1)}} \)

Step 3 Simplify

Now you simplify it by canceling out similar factors in the numerator and denominator, such as \(x+1\) and \(x+2\). The expression becomes: \( \frac{{x(x + 3)}}{{(x - 2)(x - 1)}} \).

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to break down quadratic expressions into simpler elements, which are usually products of binomials. This method is essential when working with algebraic fractions because it helps identify and cancel common factors in the numerator and denominator. For example, consider the quadratic expression \(x^2 + x\). By factoring, we rewrite it as \(x(x + 1)\). Similarly, for \(x^2 - 4\), which is a difference of squares, the factored form becomes \((x - 2)(x + 2)\). Recognizing these patterns is key in simplifying algebraic fractions and making them more manageable.
Some common techniques for factoring include:

• Finding a common factor
• Using special product formulas such as difference of squares, perfect square trinomials, and sum/difference of cubes
• Applying the quadratic formula when necessary

Remember, practice can greatly improve your factoring skills. Try to factor different quadratics regularly to gain confidence.

Rational Expressions

A rational expression is essentially a fraction where the numerator and denominator are both polynomials. Understanding rational expressions is important because they form the basis of operations in algebra, just like fractions do in basic arithmetic. For instance, in the problem \(\frac{x^2 + x}{x^2 - 4}\), you can see that both the top and bottom parts are polynomials.
When working with rational expressions:

• Always consider if the expression can be simplified or factored further
• Be mindful of any restrictions in the variables, as they dictate the values that make the denominator zero and are therefore undefined

It’s essential to approach rational expressions with the aim of simplifying them in order to make subsequent operations like addition, subtraction, multiplication, or division easier.

Multiplication of Fractions

Multiplying fractions in algebra follows the same rule as elementary arithmetic: multiply the numerators together and the denominators together. However, before you engage in multiplication, it's often useful to first simplify the fractions involved. This not only makes the calculations simpler but also ensures that the resulting expression is as simple as possible.
In the context of algebraic fractions, remember:

• Convert any division operation into a multiplication one by taking the reciprocal of the divisor. This is how \(\frac{x^2 + x}{x^2 - 4} \div \frac{x^2 - 1}{x^2 + 5x + 6}\) became \(\frac{x(x + 1)}{(x - 2)(x + 2)} \cdot \frac{(x + 2)(x + 3)}{(x - 1)(x + 1)}\).
• Once converted, cancel common factors before multiplying, to simplify the expression early on.

By simplifying first, calculations remain manageable and less prone to error.

Simplification

Simplification is the process of reducing an expression into its simplest form. In the context of algebraic fractions, it involves canceling common factors in the numerator and denominator after factoring them. This step is crucial because it not only simplifies the problem but also ensures accuracy in the final result.
Steps to effective simplification include:

• Factor both the numerator and the denominator completely
• Identify and cancel out any common factors

For example, in the simplified form \(\frac{x(x + 3)}{(x - 2)(x - 1)}\), the factors \((x + 1)\) and \((x + 2)\) were cancelled out from the original expression. Simplification makes the expression easier to work with and often reveals the core relationships within the problem.