Problem 27 - Algebra 2 and Trigonometry

Question

In \(25-30,\) perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{2 a}{a+2} \cdot \frac{a^{2}-4}{4 a^{2}} \div \frac{a-2}{a} $$

Step-by-Step Solution

Answer

The simplest form is \(\frac{1}{2a}\), undefined for \(a = -2, 0, 2\).

Step 1: Rewrite Division as Multiplication

The given expression involves division, which can be rewritten as multiplication by the reciprocal. Rewrite the problem as follows:\[\frac{2a}{a+2} \cdot \frac{a^2-4}{4a^2} \cdot \frac{a}{a-2}\]

Step 2: Simplify the Quadratic Expression

Recognize that \(a^2-4\) is a difference of squares and can be factored as \((a-2)(a+2)\). Therefore, replace \(a^2-4\) in the expression:\[\frac{2a}{a+2} \cdot \frac{(a-2)(a+2)}{4a^2} \cdot \frac{a}{a-2}\]

Step 3: Identify and Cancel Common Factors

Look for common factors in the numerator and the denominator.- \(a+2\) in the first and the second fractions can be canceled.- \(a-2\) in the second and the third fractions can be canceled.- One \(a\) in the first fraction cancels with one \(a\) in the third fraction.After cancelling, the expression simplifies to:\[\frac{2}{4a}\]

Step 4: Simplify the Result

The expression \(\frac{2}{4a}\) can be simplified by canceling a factor of 2:\[\frac{1}{2a}\]

Step 5: Identify Restrictions on the Variable

Identify the values of \(a\) for which the original expressions are undefined. The denominators cannot be zero, so identify these points:- \(a+2 = 0 \Rightarrow a = -2\)- \(4a^2 = 0 \Rightarrow a = 0\)- \(a-2 = 0 \Rightarrow a = 2\)Therefore, \(a\) cannot be \(-2\), \(0\), or \(2\).

Key Concepts

Simplifying Rational Expressions

Rational expressions are fractions that include polynomials in the numerator, the denominator, or both. Simplifying such expressions involves reducing them to their smallest form, much like simplifying numerical fractions.

To tackle this, you can follow these steps:

• Identify and factor any polynomials in the expression to reveal common components both in the numerator and the denominator. This step often paves the way for cancellation of common factors.
• Cancel out these common factors from the numerator and the denominator. This process is akin to dividing the same number from both the top and bottom of a numerical fraction.
• Ensure you don't cancel anything across the addition or subtraction; only terms multiplied together can be canceled.

For instance, in the exercise, we canceled the common factors such as \(a+2\) and \(a-2\), simplifying the expression significantly. Always remember, before simplifying, ensure the expression is correctly factored!

Polynomial Division

Division in rational expressions often involves rewriting the division as multiplication by the reciprocal of the divisor. This is a crucial step in simplifying complex rational expressions.

Here’s how you can proceed:

• Start with the original division problem. Turn the divisor (the expression you're dividing by) upside down. This makes it the reciprocal.
• Rewrite the division as a multiplication by placing the reciprocal directly after the multiplication sign.
• Follow the multiplication by performing the regular operation as usual, focusing on multiplying across numerators and denominators.

In the exercise, this method was effectively utilized by converting \(\frac{a-2}{a}\) into its reciprocal, \(\frac{a}{a-2}\), simplifying the complex division into a much more manageable multiplication problem.

Factoring Polynomials

Factoring is a critical technique in handling polynomial expressions, especially when simplifying rational expressions or solving polynomial equations.

Some common methods of factoring include:

• **Factoring out the Greatest Common Factor (GCF):** This involves identifying any common multiples that are present in all terms of the polynomial and factoring them out.
• **Difference of Squares:** Recognize patterns such as \(a^2 - b^2 = (a-b)(a+b)\), where the polynomial can be split into two binomials.
• **Factor by Grouping:** This is effective for polynomial expressions with four terms, by grouping pairs of terms and factoring each pair.

In the current exercise, the term \(a^2-4\) was factored into \((a-2)(a+2)\) using the difference of squares method. Once factored, it makes simplifying the original rational expression far simpler, as you can easily spot and cancel matching terms.