Problem 32 - Elementary Algebra

Question

Add or subtract as indicated. Be sure to express your answers in simplest forn. (Objective 1) $$\frac{n^{2}}{n-4}-\frac{16}{n-4}$$

Step-by-Step Solution

Answer

The simplified form is \( n + 4 \).

Step 1: Identify Like Terms

Look at the given expression \( \frac{n^2}{n-4} - \frac{16}{n-4} \). Both fractions have the same denominator \( n-4 \), which means they are like terms and can be combined.

Step 2: Combine Numerators

Since the denominators are the same, subtract the numerators: \( n^2 - 16 \). The combined fraction is \( \frac{n^2 - 16}{n-4} \).

Step 3: Simplify the Expression

Recognize that the numerator \( n^2 - 16 \) is a difference of squares, which can be factored as \((n - 4)(n + 4)\). Thus, the expression becomes \( \frac{(n-4)(n+4)}{n-4} \).

Step 4: Cancel Common Factors

Cancel the common factor \( n-4 \) in the numerator and the denominator: \( \frac{(n-4)(n+4)}{n-4} = n+4 \).

Step 5: State the Simplified Answer

After canceling the common factors, the simplified form of the expression is \( n + 4 \).

Key Concepts

Difference of Squares

The difference of squares is a special type of algebraic expression. It occurs when one square number is subtracted from another. The general form is \[ a^2 - b^2 \].
When you spot this, you can remember a simple factorization rule: \[ a^2 - b^2 = (a - b)(a + b) \]. This is possible because:

• The middle terms cancel out when expanded.
• You are left with only the two squares, at the start and end.

For example, consider the expression \( n^2 - 16 \). Here, \( n^2 \) is a square, and \( 16 \) is also a square since \( 16 = 4^2 \).
You can break it down as \( (n - 4)(n + 4) \), using the difference of squares method. This technique helps in simplifying complex expressions quickly and accurately.

Combining Like Terms

Combining like terms is a fundamental process in algebra that makes expressions simpler. It involves merging terms that have the same variable raised to the same power.
In the expression \( \frac{n^2}{n-4} - \frac{16}{n-4} \), both terms share the common denominator \( n-4 \).

• Because they have the same denominator, they are considered like terms.
• This allows us to combine them by subtracting the numerators.

When you subtract the numerators \( n^2 - 16 \), you are effectively merging the like terms into a single expression \( \frac{n^2 - 16}{n-4} \). This simplification makes it easier to perform further operations. Therefore, recognizing like terms helps immensely in streamlining and solving algebraic expressions.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler 'factors' that multiply to give the original polynomial. It's a crucial step in simplifying expressions and solving equations.
After combining like terms in \( \frac{n^2 - 16}{n-4} \), we identified the numerator as a difference of squares: \( n^2 - 16 \), which can be factored into \( (n - 4)(n + 4) \).

• This reveals the structure embedded in the polynomial expression.
• After factoring, common factors in the numerator and denominator can be canceled out.

In this case, the common factor \( n-4 \) in the numerator and denominator is canceled. As a result, the expression simplifies to \( n+4 \).
Mastering the skill of factoring polynomials enables efficient simplification and solution of complex algebraic problems.