91Ó°ÊÓ

Factoring polynomials is one of the fundamental skills in algebra. This process involves expressing a polynomial as a product of its factors. To factor a polynomial:

• First, look for common factors in all the terms.
• Group terms with common factors together.
• Factor out the common elements from each group.

Let's consider the numerator in our problem:
\[2xy + 2xw + y + w\]
We can group the terms and factor them as follows:
\[(2xy + 2xw) + (y + w)\]
Factor out the common terms in each group:
\[2x(y + w) + 1(y + w)\]
Now, factor out \(y + w\) from the entire expression:
\[(2x + 1)(y + w)\]
By breaking everything down into smaller steps and grouping terms strategically, we can make factoring much easier!

Simplifying a rational expression to its lowest terms means reducing it so that no common factors remain between the numerator and the denominator. Here's how you can do this:

• Factor both the numerator and the denominator completely.
• Cancel out common factors that appear in both the numerator and the denominator.

To put this into practice, let's look at our original problem:
The numerator factors to: \[(2x + 1)(y + w)\]
The denominator factors to: \[(2x + 1)(y - w)\]
Notice that \(2x + 1\) is a common factor in both the numerator and the denominator. We can cancel this out, which simplifies our expression to:
\[\frac{y + w}{y - w}\]
This is now in its lowest terms because there are no common factors left.

In a fraction or rational expression, the top part is called the numerator and the bottom part is called the denominator. To better understand their roles, consider these key points:

• The numerator reflects the number we have or the part of something we’re describing.
• The denominator indicates the total number of equal parts or the entire quantity.

For example, in a fraction like \[\frac{3x + 2}{5x - 4}\], '3x + 2' is the numerator and '5x - 4' is the denominator.
In our specific example, the original rational expression is: \[\frac{2xy + 2xw + y + w}{2xy + y - 2xw - w}\]
Through factoring and simplifying, we determined both the numerator and the denominator factors. Then, we simplified by cancelling out any common factors.
Finally, understanding the relationship between numerator and denominator helps in recognizing when cancellation is possible, leading to simpler, more manageable expressions.