Problem 91 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{2 y^{2}+11 y-1}{y+4}-3 y $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: $\frac{2y^2 + 11y - 1}{y+4} - 3y$ Answer: $\frac{-y^2 - y - 1}{y+4}$

Step 1: Identify the least common denominator (LCD)

Since the first term has a denominator of (y+4) and the second term is a monomial, we need to find a common denominator before combining them. In this case, the least common denominator (LCD) is simply the product of (y+4) and 1, which is (y+4).

Step 2: Rewrite both terms with the LCD

We need to rewrite both terms to have the same denominator. The first term is already in the correct form. For the second term, we need to rewrite it as a fraction with a denominator of (y+4). $$ -3y = \frac{-3y(y+4)}{(y+4)} $$ Now, our expression looks like this: $$ \frac{2y^2 + 11y - 1}{y+4} - \frac{3y(y+4)}{y+4} $$

Step 3: Combine the numerators

Now that both terms have the same denominator, we can combine their numerators: $$ \frac{2y^2 + 11y - 1 - 3y(y+4)}{y+4} $$

Step 4: Simplify and distribute terms in the numerator

Distribute the -3y to the terms inside the parentheses and simplify the expression: $$ \frac{2y^2 + 11y - 1 - 3y^2 - 12y}{y+4} $$ Combine like terms in the numerator: $$ \frac{-y^2 - y - 1}{y+4} $$

Step 5: Simplify the rational expression (if possible)

In this case, the numerator cannot be factored further, and as a result, we can't cancel any terms with the denominator. Thus, the expression is already in its simplest form: $$ \frac{-y^2 - y - 1}{y+4} $$

Key Concepts

Least Common Denominator

Finding the least common denominator (LCD) is an essential step when adding or subtracting rational expressions. The LCD is the smallest expression that all denominators can divide into evenly, which allows you to combine fractions with different denominators. To find the LCD, we typically factor each denominator to its prime factors and then multiply each factor the maximum number of times it occurs in any of the denominators.

In our example, with one denominator being (y+4) and the other term a monomial without a denominator, the LCD is simply (y+4) as it's the only denominator present.

• If there were more denominators, we would need to find the product of their distinct prime factors accounting for the highest power of each factor present in the denominators.

Simplifying Algebraic Expressions

When simplifying algebraic expressions, the goal is to reduce the expression to its most basic form. This involves several steps, such as combining like terms and factoring. Like terms are terms that have the exact same variable raised to the same power. To combine them, you simply add or subtract their coefficients.

In simplifying the expression \(2y^2 + 11y - 1 - 3y^2 - 12y\), we combined like terms to get the simplified numerator \( -y^2 - y - 1 \). Combining like terms is a key part of simplifying and helps to make the expression easier to work with, either for further calculations or for interpretation.

Rational Expression Simplification

The process of rational expression simplification mirrors the simplification of numeric fractions - we aim to reduce the expression to its simplest form. This includes finding and canceling common factors in the numerator and the denominator, if possible.

In our example, once we subtracted and combined like terms, we were left with \(\frac{-y^2 - y - 1}{y+4}\). To simplify further, we need to check if the numerator can be factored to reveal any common factors with the denominator. Since no such factors exist in this case, the rational expression is already in its simplest form.

• It's important to check for simplification opportunities, as neglecting to simplify fully can lead to incorrect conclusions or more complex calculations later in a problem.

Distributive Property

The distributive property allows us to multiply a single term by each term within a bracket (parentheses). According to this property, \(a(b + c) = ab + ac\).

In our example, the term -3y was distributed over (y+4), becoming \( -3y * y - 3y * 4 \), which then simplified to \( -3y^2 - 12y\). The distributive property is a vital tool in algebra, as it's used countless times for expanding expressions, simplifying equations, and even factoring.

• The correct application of the distributive property streamlines the simplification process, leading to cleaner and more accurate solutions.