Problem 30 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{9}{2 c^{2}-2 c}-\frac{5}{2 c-2} $$

Step-by-Step Solution

Answer

\( \frac{9 - 5c}{2c(c-1)} \) is the simplified result.

Step 1: Factor Denominators

First, factor the denominators of each fraction. For the first fraction: \[ 2c^2 - 2c = 2c(c-1) \] For the second fraction: \[ 2c - 2 = 2(c-1) \]

Step 2: Find Least Common Denominator (LCD)

To combine the fractions, determine the least common denominator. The LCD of \(2c(c-1)\) and \(2(c-1)\) is \(2c(c-1)\).

Step 3: Re-write Each Fraction

Re-write each fraction with the least common denominator:\[ \frac{9}{2c(c-1)} = \frac{9}{2c(c-1)} \] since it already has the LCD.For \(\frac{5}{2(c-1)}\), multiply the numerator and denominator by \(c\):\[ \frac{5}{2(c-1)} = \frac{5c}{2c(c-1)} \]

Step 4: Subtract Fractions

Now that both fractions have the same denominator, subtract them:\[ \frac{9 - 5c}{2c(c-1)} \] Subtract the numerators, keeping the denominator the same.

Step 5: Simplify, if Possible

Check if the result can be simplified further. The expression \(9 - 5c\) has no common factors with the denominator. Therefore, the expression is already in its simplest form.

Key Concepts

Factoring Denominators

Factoring denominators involves breaking down expressions into simpler components that, when multiplied together, give us the original expression. It's like finding the ingredients in a recipe. In the given exercise, the first step was to factor the denominators. Both denominators in this problem were polynomials, which often need factoring to simplify fraction operations. For example:

• The denominator of the first fraction, \(2c^2 - 2c\), was factored into \(2c(c-1)\). This was done by taking out the greatest common factor, which is \(2c\).
• Similarly, the denominator of the second fraction, \(2c - 2\), was factored as \(2(c-1)\), by factoring out the common factor, \(2\).

Factoring helps reveal the basic structure of the expression, making it easier to manipulate in solving equations. Always look for common factors that can be extracted.

Least Common Denominator

The Least Common Denominator (LCD) is crucial when dealing with the addition or subtraction of fractions that have different denominators. It's the smallest expression that each of the denominators can divide into equally. Finding the LCD allows us to rewrite each fraction to have the same denominator, which is necessary before you can add or subtract them.

• In this problem, the denominators were \(2c(c-1)\) and \(2(c-1)\). The LCD had to include all the factors present in either denominator. This resulted in \(2c(c-1)\), as \(2c(c-1)\) incorporates both \(2\) and \(c\) as well as the factor \((c-1)\).

Once the LCD is identified, fractions can be adjusted accordingly, so they share this common denominator. This ensures that operations like addition and subtraction can be performed directly.

Subtracting Fractions

Once fractions have a common denominator, you can subtract them by simply subtracting the numerators and keeping the denominator the same. This rule simplifies the operation. In the given exercise:

• The fractions \(\frac{9}{2c(c-1)}\) and \(\frac{5c}{2c(c-1)}\) both shared the least common denominator \(2c(c-1)\).
• Subtracting these involves taking the difference of their numerators, hence, \(9 - 5c\), while keeping the denominator as is.

It's important to check if the resulting expression can be simplified further. In this instance, the numerator did not share any common factors with the denominator, confirming that the fraction was already in its simplest form. Simplification is always the final step, ensuring your answer is as neat and clear as possible.