Problem 69 - A Graphical Approach to College Algebra

Question

Simplify each complex fraction. $$\frac{m-\frac{1}{m^{2}-4}}{\frac{1}{m+2}}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{m^2 - 2m - 1}{m(m-2)} \).

Step 1: Understand the Fraction Structure

We start with the complex fraction \( \frac{m-\frac{1}{m^{2}-4}}{\frac{1}{m+2}} \). This expression consists of a main numerator \( m-\frac{1}{m^{2}-4} \) and a main denominator \( \frac{1}{m+2} \). We aim to simplify the fraction by eliminating the secondary fraction components.

Step 2: Simplify the Main Denominator

The main denominator is \( \frac{1}{m+2} \). The reciprocal of this fraction is \( m+2 \), and it can be rewritten as just that since dividing by a fraction is equivalent to multiplying by its reciprocal.

Step 3: Simplify the Expression in the Numerator

The expression in the numerator is \( m-\frac{1}{m^{2}-4} \). First, note that \( m^{2}-4 \) can be factored as \( (m-2)(m+2) \). Plug the factored form back into the fraction: \( \frac{1}{m^{2}-4} = \frac{1}{(m-2)(m+2)} \).

Step 4: Combine the Numerator Expression

To combine terms in the numerator, the expression \( m - \frac{1}{(m-2)(m+2)} \) should be treated as \( \frac{m(m-2)(m+2)}{(m-2)(m+2)} - \frac{1}{(m-2)(m+2)} \) to have the same denominator. Now combine these as: \( \frac{m(m-2)(m+2) - 1}{(m-2)(m+2)} \).

Step 5: Simplify the Complex Fraction

Now write the entire expression as \( \frac{ \frac{m(m-2)(m+2) - 1}{(m-2)(m+2)} }{m+2} \) which simplifies to \( \frac{m(m-2)(m+2) - 1}{(m-2)(m+2) \cdot (m+2)} \). Cancel terms where possible such as \( (m+2) \) to get the final form: \( \frac{m(m-2) - 1}{m(m-2)} \).

Step 6: Further Simplification

Further simplify \( \frac{m(m-2) - 1}{m(m-2)} = \frac{m^2 - 2m - 1}{m(m-2)} \) by trying to simplify the expression, but since there are no common terms to factor out or cancel completely, it stays this way.

Key Concepts

Simplification of Expressions

Simplifying complex fractions might seem daunting at first, but with a step-by-step approach, it becomes much more manageable.

• First, understand the overall structure of the fraction, specifically identifying both the numerator and the denominator.
• In complex fractions, these components often include additional fractions themselves. The key is to simplify the expression by resolving these parts individually.

Breaking down a complex fraction involves resolving any nested fractions by elimination through reciprocal multiplication or finding a common denominator to combine terms effectively. An important practice in simplification is carefully substituting value expressions to unify the approach in solving. This practice not only helps in handling terms correctly but also makes further simplification steps notably easier.
Especially when dealing with algebraic terms, watch out for opportunities to cancel or combine terms to streamline the expression thoroughly.
Simplification is vital as it can reduce complex forms to more straightforward, comprehensible, and workable forms, which is crucial not only for mathematical applications but also in understanding physical relationships in applied mathematics.

Factoring Quadratics

Factoring is an essential skill in algebra, especially when working with quadratics, as it allows us to break down complex expressions into products of simpler ones. For quadratic expressions like \[ m^2 - 4 \], factoring reveals patterns that simplify further calculation. This expression is a classic difference of squares, which can be factored into \[ (m-2)(m+2) \]. Recognizing and applying these patterns helps in transforming complicated expressions and is crucial for further operations, such as division or simplification.
When factoring quadratics:

• Firstly, see if the expression can be written as a square of a term minus another square, the difference of squares.
• Always seek to transform the quadratic expression into two binomials, which can further assist in solving equations or simplifying fractions.

Mastery of factoring quadratics aids in recognizing underlying algebraic relationships, helping not only in simplification tasks but also in a variety of higher-level math problems.

Algebraic Fractions

Dealing with algebraic fractions requires understanding how to manipulate both the numerators and denominators, just like numerical fractions. Algebraic fractions extend these concepts to include variables, thereby increasing complexity slightly. In our complex fraction example, careful attention to operations such as multiplication, division, and the properties of fractions are essential. Algebraic fractions often involve:

• Simplifying expressions by canceling common factors, which is central to reducing the fraction to its simplest form.
• Using factoring techniques to reveal and eliminate like terms in both the numerator and the denominator.
• Algebraic considerations often involve finding the least common denominator especially when there are multiple fractions to combine.

Recognizing these opportunities for simplification and cancellation turns a seemingly intractable problem into a straightforward one. The objective is to handle expressions judiciously maintaining equivalence while transforming fractions into their simplest form.
Clear knowledge and skill in managing algebraic fractions are vital, as they streamline complex mathematical problems making them accessible and easier to solve.