Problem 16 - Elementary Algebra

Question

For exercises 7-32, simplify. $$ \frac{y^{2}-y}{y+7} \cdot \frac{3 y+21}{y^{2}+y} $$

Step-by-Step Solution

Answer

\frac{3(y-1)}{y+1}

Step 1 - Factorize the Numerators and Denominators

Identify common factors and simplify each part of the expression. The first numerator is \(y^2 - y\), which can be factorized as \(y(y-1)\). The second numerator \(3y + 21\) can be simplified to \(3(y + 7)\). The first denominator is \(y + 7\), which is already simplified. The second denominator \(y^2 + y\) can be factorized as \(y(y + 1)\).

Step 2 - Rewrite the Expression with Factors

Substitute the factored forms back into the original expression. This changes the expression to: \[\frac{y(y-1)}{y+7} \cdot \frac{3(y+7)}{y(y+1)}\].

Step 3 - Cancel Common Factors

Remove common factors from the numerator and denominator. First, \(y + 7\) cancels out from both the numerators and denominators, leaving \[\frac{y(y-1)}{1} \cdot \frac{3}{y(y+1)}=\frac{y(y-1) \cdot 3}{y(y+1)}\]. Next, factor out any common \(y\) terms, resulting in: \[\frac{(y-1) \cdot 3}{y+1}\].

Step 4 - Simplify the Remaining Expression

Combine remaining terms to get \[\frac{3(y-1)}{y+1}\]. This is the simplified form of the given expression.

Key Concepts

Factoring Expressions

Factoring expressions is a fundamental step in simplifying algebraic expressions. When you factor an expression, you break it down into products of simpler expressions. This process can make complicated fractions easier to manage. For instance, consider the expression in the problem:

The first numerator is given as: \( y^2 - y \), which can be written as \( y(y - 1) \).
The second numerator is: \ ( 3y + 21 ) \, which can be written as \ ( 3(y + 7) ) \.
The denominators are already given in simpler forms or can be similarly factored:

• The first denominator, \( y + 7 \), remains the same.
• The second denominator is: \ ( y^2 + y ) \, which can be written as \ ( y(y + 1) ) \.

Factoring separates the expression into components (factors) that are easier to work with.

Canceling Common Factors

Once you have factored both numerators and denominators, the next step is to cancel out any common factors that appear in both. This is an important step because it simplifies the equation further. In the given problem, after factoring, we rewrite the expression as:

\[ \frac{y(y-1)}{y+7} \cdot \frac{3(y+7)}{y(y+1)} \]

By observing, you see that \( y+7 \) appears in both a numerator and a denominator, so we can cancel them out:

\[ \frac{y(y-1)}{1} \cdot \frac{3}{y(y+1)} = \frac{y(y-1) \cdot 3}{y(y+1)} \]

Additionally, the terms \ y \ in both the numerator and denominator can be canceled out, resulting in:

\[ \frac{(y-1) \cdot 3}{y+1} \]
This process of canceling common factors is critical as it reduces the expression to its simplest form efficiently.

Rational Expressions

Rational expressions are fractions where the numerator or the denominator (or both) are polynomials. Simplifying rational expressions often involves both factoring and canceling common factors. The goal is to write the expression in its simplest form. Consider the rational expression in the problem,

\[ \frac{y^2 - y}{y + 7} \cdot \frac{3y + 21}{y^2 + y} \]

By factoring and then canceling out common factors, as explained in the previous sections, we reach the simplified expression:

\[ \frac{3(y-1)}{y+1} \]
This form of the expression is easier to interpret and work with.

Remember, whenever you're simplifying rational expressions:

• Factor both the numerator and denominator completely.
• Cancel out any common factors.
• Rewrite the simplified form.

This makes the process straightforward and ensures that you can handle more complex algebraic equations with ease.