Problem 49 - College Algebra

Question

Simplify the rational expression. $$ \frac{\frac{2 c}{c+2}+\frac{c-1}{c+1}}{\frac{2 c+1}{c+1}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3c^2 + 3c - 2}{c+2} \).

Step 1: Find a Common Denominator in the Numerator

To simplify the numerator, which is \( \frac{2c}{c+2} + \frac{c-1}{c+1} \), find a common denominator. The denominators are \( c+2 \) and \( c+1 \). Thus, the common denominator is \((c+2)(c+1)\).

Step 2: Combine the Numerator

Express each term of the numerator with the common denominator:\[ \frac{2c(c+1) + (c-1)(c+2)}{(c+2)(c+1)} \]Expand and combine the terms:\[ \frac{2c^2 + 2c + c^2 + 2c - c - 2}{(c+2)(c+1)} = \frac{3c^2 + 3c - 2}{(c+2)(c+1)} \]

Step 3: Simplify the Fraction

The original expression can now be rewritten:\[ \frac{\frac{3c^2 + 3c - 2}{(c+2)(c+1)}}{\frac{2c+1}{c+1}} \]This simplifies to:\[ \frac{3c^2 + 3c - 2}{c+2} \]

Step 4: Simplify Further if Possible

Check if \( 3c^2 + 3c - 2 \) can be factored further. Testing for common factors, there are none. Therefore, the expression in step 3 is fully simplified.

Key Concepts

Common Denominator

When simplifying rational expressions, finding a common denominator is often one of the first steps. In our exercise, the task was to simplify an expression in the numerator by combining fractions with different denominators: \( \frac{2c}{c+2} + \frac{c-1}{c+1} \).

To successfully combine these fractions, it's crucial to identify a shared baseline—this is the common denominator, which in this case is \((c+2)(c+1)\).

• The goal is to adjust the fractions so they have a common base.
• Multiply each numerator by the parts of the common denominator it lacks to achieve this.
• Doing so converts the fractions to one unified expression, enabling simplification.

Understanding this concept is fundamental in dealing with complex rational expressions, as finding a common denominator allows you to smoothly combine terms into a single fraction.

Rational Expressions

Rational expressions resemble fractions but involve polynomials in their numerators and denominators. In essence, any expression of the form \( \frac{P(x)}{Q(x)} \) where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x) eq 0\), qualifies as a rational expression.

• They play a huge role in algebraic manipulations and calculus.
• Simplifying them often requires a blend of techniques such as combining like terms, factoring, or finding common denominators.
• Despite their complexity, they follow similar rules to numerical fractions.

Through rational expressions, one can learn about division of polynomials and many algebraic identities, which are critical for more advanced mathematics.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or "factors," that when multiplied give the original polynomial. It is a crucial skill in algebra, especially when simplifying rational expressions.

• The polynomial \(3c^2 + 3c - 2\) from our exercise was checked for factorability.
• In this instance, it wasn't factorable with simple techniques, but knowing how to factor is vital for other cases.
• Common methods include finding common factors, using the distributive property, or employing special identities like the difference of squares.

By mastering factoring, students equip themselves with the tools needed to simplify and solve polynomial equations and expressions efficiently.