Problem 56 - College Algebra

Question

Perform the given operations and simplify. $$ \frac{\frac{4 a+1}{2 a-3}+\frac{2 a-3}{2 a+3}}{\frac{4 a^{2}+9}{a}} $$

Step-by-Step Solution

Answer

The simplified form involves factoring and is an intermediate result: \(\frac{a(6a^2 + a + 6)}{(2a-3)(2a+3)(4a^2+9)}\).

Step 1: Simplify the Numerator

The numerator is the sum of two fractions \( \frac{4a+1}{2a-3} + \frac{2a-3}{2a+3} \). To add these, find a common denominator, which is \((2a-3)(2a+3)\). The expression becomes \( \frac{(4a+1)(2a+3) + (2a-3)(2a-3)}{(2a-3)(2a+3)} \).

Step 2: Expand and Combine Terms in the Numerator

Expand both terms in the numerator: 1. \((4a+1)(2a+3) = 8a^2 + 12a + 2a + 3 = 8a^2 + 14a + 3\) 2. \((2a-3)(2a-3) = 4a^2 - 12a + 9\) Combine these to get: \((8a^2 + 14a + 3) + (4a^2 - 12a + 9) = 12a^2 + 2a + 12\).

Step 3: Simplify the Denominator

The denominator \(\frac{4a^2 + 9}{a}\) can be rewritten as \(\frac{4a^2}{a} + \frac{9}{a} = 4a + \frac{9}{a}\).

Step 4: Simplify the Entire Fraction

The full expression is now \(\frac{\frac{12a^2 + 2a + 12}{(2a-3)(2a+3)}}{4a + \frac{9}{a}}\). Rewriting the denominator as a single fraction, we get a common denominator \(a\): \(\frac{4a^2 + 9}{a}\). Multiply the entire expression by \(a\) to clear this out. The expression becomes \(\frac{a(12a^2 + 2a + 12)}{(2a-3)(2a+3)(4a^2+9)}\).

Step 5: Simplify Further

Notice that \(12a^2 + 2a + 12\) can be factored further. Try factoring it out: \(2(6a^2 + a + 6)\). Since no further simplification is feasible with the given steps, the final answer is simplified to the extent possible. Further simplification might not be visible without factoring out potential common terms.

Key Concepts

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions, such as polynomials. In our exercise, the algebraic fraction in the numerator, \( \frac{4a+1}{2a-3} + \frac{2a-3}{2a+3} \), consists of polynomial expressions.
When dealing with algebraic fractions, the key is to follow similar rules as you would with numeric fractions.

• Find a common denominator to combine the fractions.
• Factor and simplify wherever possible.

Understanding the properties and behavior of the variables involved will help in managing operations, such as addition or multiplication, on these algebraic fractions.

Common Denominators

Finding a common denominator is essential when adding, subtracting, or comparing algebraic fractions. It helps in merging different fractions into a single coherent expression. In the problem at hand, the common denominator was found by multiplying the individual denominators of the algebraic fractions: \((2a-3)(2a+3)\).
This process involves:

• Identifying the denominators of the fractions involved.
• Finding their least common multiple (LCM) when necessary.
• Rewriting each fraction to have this common denominator before performing operations.

Using a common denominator, we effectively align our fractions, enabling straightforward addition or subtraction. This technique is similar in concept to handling numeric fractions but extends into more complex expressions when involving variables and polynomials.

Polynomial Expansion

Polynomial expansion refers to spreading out the terms of a polynomial expression when multiplying them. It's crucial in simplifying expressions or performing operations like addition and subtraction on polynomials.
In the exercise above, we expanded \((4a+1)(2a+3)\) and \((2a-3)(2a-3)\) using the distributive property:

• Distribute each term of the first polynomial to every term of the second polynomial.
• Sum the resulting terms to form a new polynomial expression.

This ensures each combination of polynomial terms is considered, resulting in an accurate and simplified polynomial. The final result from the combined expansion was \(12a^2 + 2a + 12\), showing how these minor steps build to a broader, complete solution.

Factoring Expressions

Factoring expressions involves breaking down a complex polynomial into products of simpler polynomials. This technique allows for the simplification of expressions by identifying and extracting common factors.
In our exercise, the polynomial \(12a^2 + 2a + 12\) was factored by recognizing common numerical and algebraic components, yielding \(2(6a^2 + a + 6)\).

• Identify common factors in the polynomial terms.
• Factor these out to simplify the expression.
• Consider specific techniques such as grouping or using the quadratic formula if necessary.

Factoring can often reveal opportunities for further simplification or cancellation, especially when dealing with algebraic fractions. Even if full factorization isn't visible immediately, looking for potential patterns helps streamline future algebraic operations.