Problem 19 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{y^{2}+9 y}{y+9} $$

Step-by-Step Solution

Answer

The simplified expression is \(y\).

Step 1: Factor the numerator

Notice that the numerator is a polynomial expression. Factor out the common term in the numerator.\[ y^2 + 9y = y(y + 9) \]

Step 2: Rewrite the expression

Rewrite the original expression using the factored form obtained in step 1.\[ \frac{y(y + 9)}{y + 9} \]

Step 3: Simplify the expression

Observe that both the numerator and denominator have the term \(y + 9\). They cancel each other out, leaving: \[ y \]

Key Concepts

Factoring Polynomials

Factoring polynomials is often the first step in simplifying rational expressions. A polynomial is an algebraic expression that involves terms with variables raised to whole number exponents. When you 'factor' a polynomial, you essentially rewrite it as a product of simpler expressions.
In the given exercise, the polynomial in the numerator is \( y^2 + 9y \). This can be factored by looking for the greatest common factor (GCF).
The GCF of \( y^2 \) and \( 9y \) is \( y \). So, we rewrite the polynomial like this: \[ y^2 + 9y = y(y + 9) \]
Now the expression is prepared for the next steps in simplification. Factoring polynomials makes it easier to see common terms in both the numerator and the denominator. This is crucial for the next step: canceling terms.

Canceling Terms

Canceling terms is a straightforward but essential step in simplifying rational expressions. It involves removing common factors that appear in both the numerator and the denominator.
In the given exercise, after factoring, the expression is: \[ \frac{y(y + 9)}{y + 9} \]
The term \( y + 9 \) appears in both the numerator and the denominator. This means it can be 'canceled' or divided out.
Here's a simplified way to think about it: When you divide any number by itself, the result is 1.
So, \( \frac{y + 9}{y + 9} = 1 \).
When we apply this to our expression, we are left with \( y \). Canceling common terms makes the expression much simpler and easier to understand.

Simplification Steps

Simplification steps are the sequence of actions taken to make an expression as simple as possible. For the given exercise, let's break down these steps:

• Step 1: Factor the numerator. We factored \( y^2 + 9y \) into \( y(y + 9) \).


• Step 2: Rewrite the expression with the factored numerator: \[ \frac{y(y + 9)}{y + 9} \]


• Step 3: Cancel out the common term \( y + 9 \) in both the numerator and the denominator. We get \( y \).

Simplifying rational expressions involves recognizing these patterns and applying these steps carefully. This process can turn a complicated-looking expression into something manageable.