Problem 62 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{4 x^{2}-y^{2}}{x^{2}+4 x y+4 y^{2}}+\frac{4 x-2 y}{3 x+6 y}$$

Step-by-Step Solution

Answer

\(\frac{8 x^{2}-3 y^{2}}{3(x+2y)^{2}}\) is the final answer.

Step 1: Analyze the denominator of the first fraction

In the denominator of the first fraction, the quadratic formula could be used to simplify \(x^{2}+4 x y+4 y^{2}\), as it is in the form of \(a^{2} + 2ab + b^{2}\), which can be rewritten as \((a+b)^{2}\). Therefore, the denominator simplifies to \((x+2y)^{2}\).

Step 2: Simplify the denominator of the second fraction

In the denominator of the second fraction, the expression \(3 x+6 y\) can be simplified by taking out the common factor of 3, which results in \(3(x+2y)\).

Step 3: Replace the simplified denominators in the original problem

Substitute the simplified expressions back to the original problem. The new expression becomes: \[\frac{4 x^{2}-y^{2}}{(x+2y)^{2}}+\frac{4 x-2 y}{3(x+2y)}\]

Step 4: Find a common denominator and combine like terms

To combine the two fractions, a common denominator is needed. The denominator of the first fraction is squared and of the second is not. To equalize, multiply the numerator and denominator of the second fraction by \((x+2y)\). After multiplication, combine similar terms in the numerator. The expression becomes: \[\frac{4 x^{2}-y^{2}+4 x^{2}-2 y^{2}}{3(x+2y)^{2}} = \frac{8 x^{2}-3 y^{2}}{3(x+2y)^{2}}\]

Key Concepts

Quadratic Expressions

Quadratic expressions are mathematical phrases that involve variables raised to the power of two, typically in the form of \(ax^2 + bx + c\). When working with quadratic expressions, we often need to factor them to simplify calculations.
For instance, in our problem, the expression \(x^2 + 4xy + 4y^2\) can be viewed as a perfect square trinomial. A perfect square trinomial takes the form \(a^2 + 2ab + b^2 = (a+b)^2\).

• First, identify the terms that represent \(a\) and \(b\). In this case, \(a = x\) and \(b = 2y\).
• Rewrite the expression as \((x + 2y)^2\).

Factoring makes later processes, like division, easier by revealing common factors or patterns.

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression more manageable or easier to interpret while keeping its value unchanged.
This can involve performing operations such as combining like terms or factoring out common elements. In our exercise, let's observe how the denominators were simplified:

• The expression \(x^2 + 4xy + 4y^2\) was reduced to \((x+2y)^2\) through factoring.
• The expression \(3x + 6y\) was simplified by factoring out a common multiple, resulting in \(3(x+2y)\).

Both steps are crucial in reducing the complexity of the overall expression and helping to find a common denominator later. Simplifying not only helps you perform calculations more easily but also reduces errors and saves time during problem-solving.

Finding Common Denominators

To combine fractions, especially in polynomial division, finding a common denominator is key. This process equalizes the size of the fractions' denominators, allowing direct addition or subtraction.
The given problem involves two fractions with different denominators:

• The first: \((x+2y)^2\).
• The second: \(3(x+2y)\).

Since the denominators share a common factor \((x+2y)\), our goal is to align them by expanding the second fraction's denominator to match the first. This involves multiplying the numerator and denominator of the second fraction by \((x+2y)\), bringing it to \(3(x+2y)^2\).
This step ensures both fractions have the same denominator, allowing us to add the numerators directly. A common denominator harmonizes the terms, streamlining the process of combining fractions into a single expression.