Problem 92 - Intermediate Algebra

Question

Look Alikes . . . a. \(\frac{a^{2}-5 a+6}{2 a-4} \cdot \frac{2 a-6}{a-2}\) b. \(\frac{a^{2}-5 a+6}{2 a-4} \div \frac{2 a-6}{a-2}\)

Step-by-Step Solution

Answer

a) \(\frac{(a-3)^2}{2}\); b) \(\frac{1}{2}\).

Step 1: Factor Numerators and Denominators

Let's factor the expressions. For the numerator of both parts, we have \(a^2 - 5a + 6\), which factors into \((a-2)(a-3)\). The denominator \(2a-4\) factors into \(2(a-2)\). The expression \(2a-6\) simplifies to \(2(a-3)\).

Step 2: Simplify the Multiplication

For part (a), substitute the factored forms:\[\frac{(a-2)(a-3)}{2(a-2)} \cdot \frac{2(a-3)}{a-2} = \frac{(a-3)}{2} \cdot (a-3) = \frac{(a-3)^2}{2}\]Cancel \((a-2)\) in the numerator and denominator and simplify wherever possible.

Step 3: Simplify the Division

For part (b), division can be converted to multiplication by the reciprocal. Rewrite the division as multiplication:\[\frac{(a-2)(a-3)}{2(a-2)} \div \frac{2(a-3)}{a-2} = \frac{(a-2)(a-3)}{2(a-2)} \cdot \frac{a-2}{2(a-3)} = \frac{1}{2} \]Cancel common factors to simplify the expression.

Step 4: Verify and Conclude

For both expressions, verify the cancellation steps and ensure no division by zero (i.e., \(aeq2, 3\) to avoid undefined terms). The final simplified result for part (a) is \(\frac{(a-3)^2}{2}\) and for part (b) is \(\frac{1}{2}\).

Key Concepts

Simplifying Rational Expressions

When working with rational expressions, simplification is key. Simplifying rational expressions involves reducing the expression to its simplest form by canceling out common factors from the numerator and denominator.

To do this effectively, you must first factor the numerator and denominator whenever possible. Factoring allows you to see the common components more clearly. For instance, in the problem provided, we had the expression \( \frac{a^2 - 5a + 6}{2a - 4} \). By factoring, we obtained \( \frac{(a-2)(a-3)}{2(a-2)} \).

With both the numerator and denominator factored, you can easily identify and cancel common factors. In our example, \(a-2\) was a common factor, which allowed us to simplify this expression significantly. It’s crucial to remember not to cancel terms that are not entirely factored! Always ensure that everything is factored before canceling. Be cautious about undefined expressions, which occur when the denominator equal zero. In the example, \(aeq2\) to ensure the denominator isn't zero.

Multiplication of Rational Expressions

Multiplying rational expressions is quite similar to multiplying fractions. Here are the steps you can follow:

1. **Factor:** Like simplifying, start by factoring numerators and denominators. This allows you to easily spot and cancel common factors.

2. **Multiply Across:** Once factored, multiply across the numerators and across the denominators. In our example, before the multiplication, we had \( \frac{(a-2)(a-3)}{2(a-2)} \cdot \frac{2(a-3)}{a-2} \).

3. **Cancel Out Matching Factors:** With everything factored, you can cancel out any common factors present in both the numerators and denominators. Here, \(a-2\) cancels out immediately.

4. **Simplify Further:** After cancelling, if there are any other simplifications possible, carry them out. We ended with \( \frac{(a-3)^2}{2} \) as the simplified result for the multiplication in our example.

Division of Rational Expressions

Division of rational expressions might seem challenging, but it becomes easier when you use the reciprocal. Follow these steps for successful division:

1. **Convert to Multiplication:** Turn the division into multiplication by using the reciprocal of the divisor. For example, change \( \frac{(a-2)(a-3)}{2(a-2)} \div \frac{2(a-3)}{a-2} \) into \( \frac{(a-2)(a-3)}{2(a-2)} \cdot \frac{a-2}{2(a-3)}\).

2. **Follow Multiplication Steps:** Factor all parts of the expressions first to identify what can be canceled.

3. **Cancel Like Terms:** Once factored, cancel any common terms in the numerators and denominators across the entire expression. This simplification led us to \( \frac{1}{2} \) in the final division result.

4. **Check for Undefined Values:** Remember, just as with multiplication, ensure the values that make the denominator zero are noted to avoid undefined expressions, ensuring a clear valid result.