Problem 40 - College Algebra with Corequisite Support

Question

For the following exercises, add and subtract the rational expressions, and then simplify. \(\frac{4 p}{p+1}-\frac{p+1}{4 p}\)

Step-by-Step Solution

Answer

\(\frac{15p^2 - 2p - 1}{4p(p+1)}\)

Step 1: Identify a Common Denominator

To add or subtract rational expressions, we need a common denominator. The denominators given are \(p+1\) and \(4p\). The common denominator will be the product of these two expressions: \((p+1) \times (4p) = 4p(p+1)\).

Step 2: Rewrite With a Common Denominator

Rewrite each fraction using the common denominator \(4p(p+1)\). For the first expression, \(\frac{4p}{p+1}\) becomes \(\frac{4p \times 4p}{4p(p+1)} = \frac{16p^2}{4p(p+1)}\). Similarly, for the second expression \(\frac{p+1}{4p}\), multiply both the numerator and the denominator by \(p+1\): \(\frac{(p+1)(p+1)}{4p(p+1)} = \frac{(p+1)^2}{4p(p+1)}\).

Step 3: Subtract the Numerators

Now that both fractions have the same denominator, subtract the numerators: \(\frac{16p^2}{4p(p+1)} - \frac{(p+1)^2}{4p(p+1)}\). This simplifies to \(\frac{16p^2 - (p^2 + 2p + 1)}{4p(p+1)}\).

Step 4: Simplify the Numerator

Simplify the expression in the numerator by distributing and combining like terms: \(16p^2 - (p^2 + 2p + 1) = 16p^2 - p^2 - 2p - 1\). This simplifies to \(15p^2 - 2p - 1\).

Step 5: Simplify the Expression

The simplified form is \(\frac{15p^2 - 2p - 1}{4p(p+1)}\). Check if the numerator can be factored further for simplification. Since \(15p^2 - 2p - 1\) is already in its simplified form with no further common factors with the denominator, this is the final simplified expression.

Key Concepts

Common Denominator

One of the essential steps when working with rational expressions involves finding a common denominator. This process is much like what you do with regular fractions. In a fraction, the denominator tells you into how many parts the whole is divided. To add or subtract fractions, they need to have the same denominator because it ensures you're working with the same size parts.
In our problem, we first notice the denominators are different: one is \(p+1\) and the other is \(4p\). We need a common denominator to proceed, and this will be the product of these two: \((p+1) \times 4p = 4p(p+1)\). This new expression represents a shared baseline, allowing the two rational expressions to be combined.
Think of the common denominator as a common language that both fractions can speak. Once you have this common basis, you can write both expressions as parts of the same whole, enabling you to add or subtract them effectively.

Subtracting Fractions

Subtracting fractions is another crucial technique in dealing with rational expressions. Once you have a common denominator, you can directly subtract the numerators, just like you do with simple numeric fractions.
In the given problem,

• the first fraction \(\frac{4p}{p+1}\) can be rewritten with the common denominator as \(\frac{16p^2}{4p(p+1)}\).
• The second fraction, \(\frac{p+1}{4p}\), becomes \(\frac{(p+1)^2}{4p(p+1)}\).

Once you have these representations, subtraction happens at the numerators: \(16p^2 - (p+1)^2\). Notice, both expressions share the same denominator, so we merely focus on subtracting the numerators smoothly, simplifying the expression further. This step often involves a bit of careful algebra, especially combining like terms and managing any negative signs during the subtraction.

Simplifying Algebraic Expressions

After subtracting the numerators, simplifying the resulting algebraic expression comes next. This involves distributing any negative signs and combining like terms, which are terms that have the same variable raised to the same power.
In our exercise, after subtracting, we get \(16p^2 - (p^2 + 2p + 1)\), which simplifies to \(15p^2 - 2p - 1\). Simplifying means making the expression as neat as possible without changing its value.

• Distribute and simplify all terms inside parentheses.
• Combine like terms to get a compact and reduced form of the expression, if possible.

The process might also involve checking if the expression can be factored further. In our case, the expression was already as simplified as it could go. Simplifying algebraic expressions aids in understanding potential further operations you may perform on them and results in a tidier, more manageable form.