Problem 17 - Algebra and Trigonometry

Question

Simplify the rational expression. $$ \frac{5 y^{2}}{10 y+y^{2}} $$

Step-by-Step Solution

Answer

Simplified expression is \(\frac{5y}{y + 10}\).

Step 1: Factor the numerator and the denominator

First, observe that the numerator is already factored as \(5y^2\). Now, let's factor the denominator. The denominator is \(10y + y^2\), which can be written as \(y^2 + 10y\). We can factor out a \(y\) from both terms, giving us \(y(y + 10)\).

Step 2: Write the expression with factored numerator and denominator

The expression with the factored numerator and denominator is now \(\frac{5y^2}{y(y + 10)}\). This shows us that there are common factors present.

Step 3: Cancel the common factor

Notice that both the numerator and the denominator have a common factor of \(y\). Cancel this factor, leaving you with \(\frac{5y}{y + 10}\).

Step 4: Verify the simplification

Check that the expression \(\frac{5y}{y + 10}\) cannot be simplified further by confirming there are no other common factors.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step in simplifying rational expressions. It involves breaking down a polynomial into simpler terms, known as factors, that can be multiplied together to give the original polynomial.

For instance, if you look at the expression \(10y + y^2\), we can rearrange it as \(y^2 + 10y\). Both terms here share a common factor, which is \(y\).

Therefore, by factoring out a \(y\), we end up with \(y(y + 10)\).

This process allows us to find factors that can be canceled with those in the numerator, making the entire expression simpler. In essence, factoring helps expose the underlying structure of the polynomial, making it easier to work with in further simplification steps.

Canceling Common Factors

Canceling common factors is a method used to simplify expressions by reducing fractions. Once the numerator and denominator are both factored, we can identify any common factors that appear in both.

In the expression \(\frac{5y^2}{y(y + 10)}\), the common factor here is \(y\).

Since \(y\) appears both in the numerator \(5y^2\) and the denominator \(y(y + 10)\), we can divide both parts by \(y\).

This leaves us with the simplified expression \(\frac{5y}{y + 10}\).

By canceling these common factors, you simplify the expression, making it more straightforward and manageable.

Numerator and Denominator

In a rational expression like \(\frac{a}{b}\), the numerator is the top part \(a\) and the denominator is the bottom part \(b\). Understanding the roles of the numerator and denominator is essential in simplifying rational expressions.

The goal is often to make these expressions simpler by factoring and canceling common terms.

In our example, the numerator starts as \(5y^2\). It is already factored as fully as needed; however, in other cases, further factoring might be required.

The denominator initially is \(10y + y^2\), but it can be factored down to \(y(y + 10)\).

By simplifying each part carefully, we are able to identify opportunities to reduce the rational expression to its simplest form.