Problem 70 - A Graphical Approach to College Algebra

Question

Simplify each complex fraction. $$\frac{\frac{3}{p^{2}-16}+p}{\frac{1}{p-4}}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{3 + p^3 - 16p}{p+4} \).

Step 1: Identify the Structure

The complex fraction involves a fraction within a fraction. The numerator is \( \frac{3}{p^2 - 16} + p \) and the denominator is \( \frac{1}{p-4} \). We need to simplify this by eliminating these inner fractions.

Step 2: Simplify the Numerator

The numerator \( \frac{3}{p^2 - 16} + p \) can be combined by finding a common denominator. Notice that \( p^2 - 16 \) is a difference of squares: \( (p-4)(p+4) \). Thus, the common denominator will be \( (p-4)(p+4) \).

Step 3: Rewrite the Numerator with Common Denominator

Rewrite the term \( p \) as \( \frac{p(p^2-16)}{(p-4)(p+4)} \) to have a common denominator. Thus, the numerator becomes:\[\frac{3}{(p-4)(p+4)} + \frac{p(p^2-16)}{(p-4)(p+4)} = \frac{3 + p(p^2 - 16)}{(p-4)(p+4)}\]

Step 4: Simplify the Denominator

The denominator is \( \frac{1}{p-4} \), which can be rewritten over a common denominator. In this case, the common denominator is \( p-4 \).

Step 5: Multiply by Reciprocal

To simplify the complex fraction, multiply the simplified numerator by the reciprocal of the denominator:\[\frac{3 + p(p^2 - 16)}{(p-4)(p+4)} \times (p-4) = \frac{3 + p(p^2 - 16)}{p+4}\]

Step 6: Simplify Final Expression

Now simplify \( 3 + p(p^2 - 16) \). Since \( p^2 - 16 = (p-4)(p+4) \), substitute back or simply expand and combine terms. This expression simplifies to a polynomial expression using algebraic rules. Verify and expand: \( 3 + p^3 - 16p \).

Step 7: Combine and Simplify Further

Combine like terms to produce a final polynomial expression over \( p+4 \). No further simplification inside the fraction is possible unless factoring reveals further common factors.

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a process where we reduce expressions to their simplest form, making them easier to work with and understand. This process often involves combining like terms, factoring, or canceling common factors.
Let's consider an example: Suppose we have a complex fraction like \( \frac{\frac{3}{p^2-16}+p}{\frac{1}{p-4}} \). The goal is to simplify this expression by removing fractions within fractions, making it more manageable.

• First, rewrite any terms so that they share a common denominator.
• Then, identify and combine any like terms.
• Finally, eliminate any remaining complex components by multiplying by reciprocal, which simplifies the fraction entirely.

Simplifying expressions helps solve equations, analyze functions, and work through algebra problems with greater efficiency and clarity. It's a fundamental skill in algebra that forms the basis of much more complex mathematics.

Polynomials

Polynomials are expressions composed of variables and coefficients, structured with operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. They're vital in algebra because they model various phenomena and appear in equations everywhere.
A polynomial can be something as simple as \( p \) or as complex as \( p^3 - 16p + 3 \), comprising several terms.

• Each term consists of a coefficient (the numerical part) and a variable raised to an exponent.
• For instance, in \( 3p^2 \), the number 3 is the coefficient, and \( p^2 \) represents a variable \( p \) raised to the power of 2.

Operations on polynomials include addition, subtraction, multiplication, and sometimes division by simple factors. Understanding how to manipulate these expressions is crucial when simplifying complex fractions or resolving algebraic equations. Recognize patterns such as the difference of squares, like \( p^2 - 16 = (p-4)(p+4) \), to factor and simplify effectively.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They are a crucial aspect of algebra, allowing us to model ratios or fractional parts in calculations.
For instance, an expression like \( \frac{3 + p(p^2 - 16)}{p+4} \) represents a rational expression, as both the top and bottom are polynomial terms.

• The key to working with rational expressions is simplifying them to their lowest terms by factoring and canceling common factors.
• Understand the restrictions on the variables where the denominator can't equal zero to prevent undefined expressions.

Simplifying rational expressions often involves rewriting them using common denominators or multiplying by the reciprocal of another expression to eliminate complex fractions. By understanding these principles, managing both simple and complex algebraic problems becomes easier. Mastery of rational expressions enables you to engage more deeply with equations and functions in algebra.