Problem 24 - Algebra 2 and Trigonometry

Question

In \(13-24,\) divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \left(a^{2}-1\right) \div \frac{2 a+2}{a} $$

Step-by-Step Solution

Answer

Simplified quotient: \( \frac{a(a-1)}{2} \); Undefined for \( a = 0 \) and \( a = -1 \).

Step 1: Interpret Division as Multiplication

To divide by a fraction, we multiply by its reciprocal. Thus, \( (a^2 - 1) \div \frac{2a + 2}{a} \) becomes \( (a^2 - 1) \times \frac{a}{2a + 2} \).

Step 2: Factorize the Numerators and Denominators

The numerator \( a^2 - 1 \) can be factored using the difference of squares as \((a-1)(a+1)\). Similarly, the denominator \( 2a + 2 \) can be factored to \( 2(a + 1) \). Therefore, the expression becomes \( (a - 1)(a + 1) \times \frac{a}{2(a + 1)} \).

Step 3: Simplify the Expression

Cancel the common factors in the numerator and the denominator. The term \((a+1)\) appears in both the numerator and the denominator, so they cancel out. The expression simplifies to \( \frac{(a - 1) \cdot a}{2} \) or \( \frac{a(a - 1)}{2} \).

Step 4: Identify Undefined Values

A fraction is undefined where the denominator equals zero. The original fraction \( \frac{2a + 2}{a} \) is undefined when \( a = 0 \) and also where \( a+1 = 0 \), which results in \( a = -1 \). Therefore, the expression is undefined for \( a = 0 \) and \( a = -1 \).

Key Concepts

Division of Fractions

Dividing fractions might seem tricky at first, but with the right approach, it becomes quite manageable. The key idea is that dividing by a fraction is the same as multiplying by its reciprocal.

Let's say you have to divide fraction A by fraction B. Instead of dividing, you flip fraction B, turning the division into multiplication. For example, to solve \( \frac{a}{b} \div \frac{c}{d} \), you convert it to \( \frac{a}{b} \times \frac{d}{c} \). This makes things a lot easier!

Notice that the operation switches from "divide" to "times," and the second fraction (B) is flipped upside down. This simple trick will help you tackle any fraction division problem.

• Change "division" to "multiplication."
• Flip the second fraction upside down (find its reciprocal).
• Solve as a multiplication problem.

Simplifying Rational Expressions

Rational expressions are fractions that have polynomials as their numerators and denominators. Simplifying these expressions involves reducing the fraction to its simplest form, just like with numerical fractions.

Here’s how you do it: you look for factors that are common to both the numerator and the denominator and cancel them out. This is similar to reducing the fraction \( \frac{4}{6} \) to \( \frac{2}{3} \) by canceling a common factor (in this case, 2).

Take \( \frac{(a-1)(a+1)}{2(a+1)} \) for example: both the numerator and the denominator contain \( (a+1) \), so they cancel each other out. This leaves us with the expression \( \frac{a(a-1)}{2} \).

• Factor both the numerator and the denominator, if possible.
• Identify and cancel any common factors.
• Write the expression in its simplest form.

Undefined Values in Fractions

A crucial concept in working with any fraction is understanding when it is undefined. Fractions are undefined whenever their denominator is equal to zero because division by zero is impossible.

Let’s examine the fraction \( \frac{m}{n} \). For this fraction to be valid, \( n eq 0 \). Hence, we must identify the values that make the denominator zero to understand where our expression doesn’t make sense.

In the example, \( \frac{2a + 2}{a} \) is undefined when \( 2a + 2 = 0 \) or \( a = -1 \) and when \( a = 0 \). These are the values of \( a \) that make the denominator zero.

Remember this: if at any step a denominator could potentially become zero, pause and determine those problematic values to ensure your expressions remain meaningful.

• Set the denominator equal to zero to find undefined values.
• Solve the resulting equation.
• Note these values as points where the fraction cannot exist.