Problem 32 - Elementary Algebra

Question

Simplify each algebraic fraction. $$\frac{n^{2}+9 n+18}{n^{2}+6 n}$$

Step-by-Step Solution

Answer

The simplified form of the algebraic fraction is \( \frac{n+3}{n} \).

Step 1: Identify the Numerator and Denominator

In the exercise, the algebraic fraction to be simplified is \( \frac{n^2 + 9n + 18}{n^2 + 6n} \). Here, the numerator is \( n^2 + 9n + 18 \) and the denominator is \( n^2 + 6n \).

Step 2: Factor the Numerator

Look for two numbers whose product is the constant term (18) and whose sum is the coefficient of the middle term (9). These numbers are 3 and 6. Therefore, the numerator can be factored as \( (n+3)(n+6) \).

Step 3: Factor the Denominator

Factor the denominator \( n^2 + 6n \) by taking out the greatest common factor, which is \( n \). So, the denominator becomes \( n(n+6) \).

Step 4: Simplify the Fraction by Canceling Common Factors

The factored form of the fraction is \( \frac{(n+3)(n+6)}{n(n+6)} \). Since \( n+6 \) is a common factor in both the numerator and the denominator, it can be canceled out, resulting in \( \frac{n+3}{n} \).

Key Concepts

Factorization

Factorization is a critical step in simplifying algebraic fractions. It involves breaking down complex algebraic expressions into simpler multiplicative factors. This makes it easier to identify and cancel common factors later.

To factor the given expression in our exercise, start with the numerator:

• Look for numbers that when multiplied give the constant term, and when added, give the coefficient of the middle term.
• In the example of the numerator \( n^2 + 9n + 18 \), the numbers are 3 and 6 because \( 3 \times 6 = 18 \) and \( 3 + 6 = 9 \).
• This means the numerator can be expressed in its factored form as \((n+3)(n+6)\).

The denominator \( n^2 + 6n \) is factored by extracting the common factor, which is \( n \). So, it becomes \( n(n+6) \). Factorization helps uncover these elements clearly so that the next steps, such as simplification, become straightforward.

Simplification

Simplification refers to the process of reducing expressions to their simplest form. This often involves canceling out common factors between the numerator and the denominator in algebraic fractions.

Once the expressions are factored, as shown in the numerically factored form \( \frac{(n+3)(n+6)}{n(n+6)} \), simplification becomes much more straightforward.

• Identify identical expressions in the numerator and denominator.
• Here, \( n+6 \) appears in both the numerator and the denominator.
• Cancel the common factor \( n+6 \) from both the numerator and the denominator.

Post-cancellation, the expression is simplified to \( \frac{n+3}{n} \). The simplified form retains the properties of the original fraction but is often more manageable in solving equations or performing further mathematical operations.

Common Factors

Finding common factors is essential in reducing algebraic fractions effectively. A common factor is a value that can divide two or more terms without leaving a remainder.

In the given exercise, identifying \( n+6 \) as a common factor was a pivotal step:

• Common factors are extracted to reveal shared components of the numerator and denominator.
• This discovery allows for simplification, as shown when \( (n+6) \) is canceled from both parts of the fraction \( \frac{(n+3)(n+6)}{n(n+6)} \).
• The rest of the expression, \( \frac{n+3}{n} \), is free from removable common factors and is thus in its simplest form.

Throughout algebra, the ability to locate and cancel common factors simplifies expressions and highlights direct relationships between terms, facilitating clarity and further manipulations.