Problem 10 - Algebra for College Students

Question

Perform the indicated operations, and express your answers in simplest form. $$ \frac{3 n}{n^{2}-36}-\frac{2}{5 n+30} $$

Step-by-Step Solution

Answer

\( \frac{13n + 12}{5(n+6)(n-6)} \)

Step 1: Identify a common denominator

To subtract these fractions, we must have a common denominator. The denominators are of the fractions \( n^2 - 36 \) and \( 5n + 30 \). Notice that \( n^2 - 36 \) can be factored as \( (n + 6)(n - 6) \) and \( 5n + 30 \) can be factored as \( 5(n + 6) \). The least common denominator (LCD) is \( 5(n + 6)(n - 6) \).

Step 2: Rewrite each fraction with the common denominator

For \( \frac{3n}{n^2 - 36} \), multiply the numerator and the denominator by 5 to make the denominator \( 5(n + 6)(n - 6) \). This results in \( \frac{15n}{5(n+6)(n-6)} \).For \( \frac{2}{5n + 30} \), multiply the numerator and denominator by \( n - 6 \) to make the denominator \( 5(n + 6)(n - 6) \). This results in \( \frac{2(n-6)}{5(n+6)(n-6)} \).

Step 3: Subtract the numerators

Now that both fractions have the same denominator, subtract the numerators:\[ \frac{15n}{5(n+6)(n-6)} - \frac{2(n-6)}{5(n+6)(n-6)} = \frac{15n - 2(n-6)}{5(n+6)(n-6)} \]Simplify the numerator by distributing the \(-2\):\[ 15n - 2n + 12 = 13n + 12 \]

Step 4: Write the final expression in simplest form

The expression \( \frac{13n + 12}{5(n+6)(n-6)} \) is in its simplest form because the numerator \( 13n + 12 \) and the denominator \( 5(n + 6)(n - 6) \) share no common factors.Thus, the answer is:\[ \frac{13n + 12}{5(n+6)(n-6)} \]

Key Concepts

Simplifying Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. Simplifying these expressions means reducing them to their simplest form, where they can no longer be reduced without changing their value.
To simplify a rational expression, the first step is to fully factor both the numerator and the denominator. Look for common factors that can cancel each other out. This helps reduce the fraction, much like simplifying basic numerical fractions. Cancel common factors carefully, ensuring that you only cancel terms that appear exactly in both the numerator and the denominator.
It’s important to remember that you can only simplify when variables are not equal to any value that makes the original denominator zero. This is because division by zero is undefined and can lead to incorrect simplifications.

• First, check if the numerator and denominator have any common factors.
• Factor them completely.
• Cancel out common factors.

With practice, you’ll become faster at spotting opportunities to simplify rational expressions effectively.

Least Common Denominator

Finding the least common denominator (LCD) is key when you want to add or subtract fractions, including rational expressions. The common denominator is a necessary step because fractions need to have the same denominator to be combined.
For rational expressions, the least common denominator is determined by factoring the denominators of all the fractions involved. Once factored, the LCD is the least that contains all factors from each denominator involved, taken to their highest power.
In our exercise, the denominators were factored as \( (n + 6)(n - 6) \) and \( 5(n + 6) \). So, the LCD became \( 5(n + 6)(n - 6) \). It includes all factors and ensures that the simplified form of combined fractions remains uniform.

• Factor each denominator completely.
• Include each factor the largest number of times it appears in any one denominator.
• This forms your least common denominator.

Once you have the LCD, rewrite each fraction by adjusting the numerator through multiplication, so that both denominators are just the LCD.

Factoring Polynomials

Factoring polynomials is crucial for simplifying rational expressions and finding the least common denominator. Factors of a polynomial expression are expressions you can multiply to get back the original polynomial.
A common method is to check if the polynomial is a difference of squares, which was used in our task. The difference of squares has the format \( a^2 - b^2 = (a + b)(a - b) \).
In the example, \( n^2 - 36 \) was factored as \( (n + 6)(n - 6) \), where \( n^2 \) and \( 36 \) are both perfect squares. Similarly, \( 5n + 30 \) was factored into \( 5(n + 6) \), by factoring out the greatest common factor which was 5.

• Look for the greatest common factor.
• Check for patterns like the difference of squares.
• Factor out common variables if possible.

By mastering these techniques, you’ll be able to handle more complex polynomials and rational expressions with ease.

Subtracting Fractions

Before rational expressions can be subtracted, their denominators must match. Once a common denominator is established, subtracting involves three main steps.
1. **Setup:** Rewrite each fraction using the LCD as the new denominator by adjusting each fraction’s numerator properly.
2. **Combine:** With the common denominator in place, subtract the numerators only, leaving the denominator unchanged.
In our exercise, this was done by ensuring both fractions adjusted to the common denominator of \( 5(n + 6)(n - 6) \) before subtracting the numerators \( 15n \) and \( 2(n-6) \).
Finally, simplify by combining like terms in the resulting numerator. This meant expanding and simplifying \( 15n - 2(n-6) = 13n + 12 \).

• Ensure both fractions have the same denominator.
• Subtract numerators directly.
• Simplify the resulting expression.

This way, subtracting fractions becomes a straightforward process and fits neatly into managing rational expressions.