Problem 55 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y^{2}+2 y+1}+\frac{4}{y^{2}+5 y+4}$$

Step-by-Step Solution

Answer

The solution to the given problem is \(\frac{y^3+9y^2+12y+4}{(y^{2}+2y+1)\cdot(y^{2}+5y+4)}\).

Step 1: Finding the Common Denominator

In order to add two fractions, they must have the same denominator. The denominators for the given fractions are \(y^{2}+2y+1\) and \(y^{2}+5y+4\). The common denominator for these fractions is simply the product of these two. Thus, the common denominator is \((y^{2}+2y+1)(y^{2}+5y+4)\).

Step 2: Rewriting the Fractions with the Common Denominator

Before we add the fractions, each fraction has to be rewritten with the common denominator. Multiply the first fraction by \((y^{2}+5y+4)/(y^{2}+5y+4)\) and the second fraction by \((y^{2}+2y+1)/(y^{2}+2y+1)\). Doing this, we get the equivalent fractions: \(\frac{y\cdot(y^{2}+5y+4)}{(y^{2}+2y+1)\cdot(y^{2}+5y+4)} + \frac{4\cdot(y^{2}+2y+1)}{(y^{2}+2y+1)\cdot(y^{2}+5y+4)}\).

Step 3: Simplifying the Numerators

Simplify the numerators in both fractions by distributing the multiplication. The first fraction becomes \(\frac{y^3+5y^2+4y}{(y^{2}+2y+1)\cdot(y^{2}+5y+4)}\) and the second fraction becomes \(\frac{4y^2+8y+4}{(y^{2}+2y+1)\cdot(y^{2}+5y+4)}\).

Step 4: Adding the Fractions

Now that the fractions have the same denominator, they can be added. Adding the fractions gives \(\frac{y^3+5y^2+4y + 4y^2+8y+4}{(y^{2}+2y+1)\cdot(y^{2}+5y+4)}\).

Step 5: Simplifying the Resulting Fraction

Simplify the resulting fraction by combining like terms in the numerator to obtain the final answer. The resulting fraction is \(\frac{y^3+9y^2+12y+4}{(y^{2}+2y+1)\cdot(y^{2}+5y+4)}\).