Problem 9 - Algebra and Trigonometry

Question

\(7-20=\) Simplify the rational expression. $$ \frac{5 y^{2}}{10 y+y^{2}} $$

Step-by-Step Solution

Answer

The simplified rational expression is \(\frac{5y}{10+y}\).

Step 1: Factor the Numerator and Denominator

First, we need to factor both the numerator and the denominator if possible. The numerator \(5y^2\) is already factored. Now let's look at the denominator, \(10y + y^2\). We factor out the greatest common factor, which is \(y\), so it becomes \(y(10 + y)\).

Step 2: Simplifying the Expression

Now that both the numerator and the denominator are factored, the expression is \(\frac{5y^2}{y(10+y)}\). We can cancel out one \(y\) from the numerator with the \(y\) in the denominator. This simplification gives you \(\frac{5y}{10+y}\).

Step 3: Final Simplified Rational Expression

Since there are no more common factors left to cancel, the simplified form of the rational expression is \(\frac{5y}{10+y}\).

Key Concepts

Factoring

Factoring is one of the core concepts in understanding rational expressions and simplifying them. It involves breaking down expressions into simpler components or factors. Think of it like dividing a mission into smaller parts to make it easier to handle. In many cases, expressions can be rewritten as a product of their factors. For instance:

• The expression \(5y^2\) in our example is already factored. It consists of the number 5 and the variable \(y\), squared.
• For the denominator \(10y + y^2\), we need to find its factors. This could involve finding common terms or using special identities to rewrite it.

In our step-by-step solution, we factored the denominator by taking out the common variable \(y\). Thus, \(10y + y^2\) became \(y(10 + y)\). Factoring becomes especially useful when simplifying expressions because it reveals potential elements that can be canceled out later.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. In the context of simplifying rational expressions, calculating the GCF allows us to factor out the greatest possibility common to each term. This makes it simpler to find and eliminate common factors. Here's how you can easily identify the GCF:

• Look at the coefficients (numerical parts) of terms and identify the largest number that can divide them.
• Identify variables that are common to both terms, including their lowest power.

In our exercise, the terms in the denominator are \(10y\) and \(y^2\). By examining these terms:

• The numerical part 10 has a number factor of 10, and \(y^2\) shares the variable \(y\).
• The common factor is \(y\), which can be extracted from both terms.

By taking out \(y\), the expression \(10y + y^2\) simplifies to \(y(10 + y)\). This process is fundamental to reaching an expression's simplest form.

Canceling Common Factors

Canceling common factors is a crucial step in simplifying rational expressions. Before canceling, ensure that factors are present both in the numerator and the denominator. It's much like balancing a scale by removing equal weights from both sides. In our expression:

• After factoring, the expression \[\frac{5y^2}{y(10+y)}\] allows us to see a common \(y\) term that can be canceled.
• The common factor \(y\) is present both in the numerator and denominator.

By removing the shared factor \(y\), we simplify the expression further to \[\frac{5y}{10+y}\]. This is the most streamlined form achievable for this expression, as no more common factors remain to be canceled.
Always remember, cancels can only occur when a factor is multiplied on both sides. Keep expressions neat and always double-check that the factors you're canceling are indeed in product form.