Problem 72 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{y-2}{y^{2}+6 y}+\frac{y+4}{y^{2}+5 y-6} $$

Step-by-Step Solution

Answer

Question: Add the following rational expressions and simplify the result: $$ \frac{y-2}{y^2 + 6y} + \frac{y+4}{y^2 + 5y - 6} $$ Answer: $$ \frac{2y^2 + y + 2}{y(y-1)(y+6)} $$

Step 1: Factor the denominators

We start by factoring the denominators of both fractions. $$ y^2 + 6y = y(y+6) $$ $$ y^2 + 5y - 6 = (y-1)(y+6) $$

Step 2: Determine the least common denominator

Next, we determine the least common denominator (LCD) by finding the least common multiple (LCM) of the two denominators. The LCM of the two denominators is the product of all factors that appear in either denominator, raised to the highest power they appear in either denominator. In this case, the LCM of \((y(y+6))\) and \(((y-1)(y+6))\) is \((y(y-1)(y+6))\). Therefore, the LCD is \((y(y-1)(y+6))\).

Step 3: Normalize the fractions to the least common denominator

To add the fractions, we first convert them to fractions with the LCD as the denominator by multiplying the numerator and denominator of each fraction by the appropriate factor. $$ \frac{y-2}{y(y+6)} \cdot \frac{y-1}{y-1} = \frac{(y-2)(y-1)}{(y(y-1)(y+6))} $$ $$ \frac{y+4}{(y-1)(y+6)} \cdot \frac{y}{y} = \frac{y(y+4)}{(y(y-1)(y+6))} $$

Step 4: Add the normalized fractions

Now that both fractions have the common denominator, we go ahead and add the numerators and keep the denominator the same. $$ \frac{(y-2)(y-1)}{y(y-1)(y+6)} + \frac{y(y+4)}{y(y-1)(y+6)} = \frac{(y-2)(y-1) + y(y+4)}{y(y-1)(y+6)} $$

Step 5: Simplify the expression

Finally, we simplify the expression by expanding the numerators and combining like terms. $$ \frac{y^2 - 3y + 2 + y^2 + 4y}{y(y-1)(y+6)} = \frac{2y^2 + y + 2}{y(y-1)(y+6)} $$ So the final answer is: $$ \frac{2y^2 + y + 2}{y(y-1)(y+6)} $$

Key Concepts

Factoring Polynomials

Factoring polynomials is akin to breaking down numbers into their prime components, but in this case, we deal with algebraic expressions. It's a method used to express a polynomial as the product of simpler polynomials (called factors) that, when multiplied, will give back the original polynomial. This process is essential because it simplifies complex algebraic operations, like adding, subtracting, or finding the roots of polynomials.

Understanding how to factor is key especially when working with rational expressions, as we typically need to factor the denominators for finding the least common denominator (LCD) or simplifying the expression. Let’s demonstrate with a basic guideline:

• Firstly, look for a common factor in all terms of the polynomial.
• If the polynomial has three terms, identify if it's a trinomial which can be factored into a product of two binomials.
• If there are four or more terms, attempt to group them and factor by grouping.
• Special patterns like difference of squares, cubes, or perfect square trinomials can be applied.

In the exercise given, factoring the denominators is the foundation from which we can build our solution. The first denominator factors into a single binomial, while the second one factors into two binomials.

Least Common Denominator

When adding or subtracting fractions, the least common denominator (LCD) is crucial. It's the smallest expression that can be used as a common denominator for all the fractions involved. In essence, it's the least common multiple (LCM) of the denominators.

Finding the LCD ensures that fractions can be combined without altering their values. To find the LCD:

• List the factors of each denominator.
• Include each factor the greatest number of times it occurs in any of the denominators.
• If there are common factors, don't repeat them; instead, use the highest power among the denominators.

For the task at hand, the LCD is found by recognizing that both denominators share a \(y+6\) term. We take the unique factors from each denominator and multiply them to form the LCD.

Simplifying Algebraic Fractions

To simplify an algebraic fraction, we seek to minimize its complexity, making it easier to understand or further manipulate. This is commonly done after operations like addition, subtraction, multiplication, or division of fractions have taken place.

Here are the steps to simplify an algebraic fraction:

• Factor both the numerator and the denominator as completely as possible.
• Identify and eliminate common factors in the numerator and the denominator.
• Apply the distributive property to combine like terms, if necessary.

Our exercise involved adding two fractions with different denominators, which requires a common denominator. After normalizing the fractions and combining them, we are left with an expression that might look complex. However, we can simplify it by expanding the numerators and then combining like terms which are similar to adding or subtracting coefficients in similar algebraic terms.