91Ó°ÊÓ

Factoring by grouping is a technique used to simplify polynomials by grouping terms with common factors.
This method works best when the polynomial has four or more terms.
To start, you group the terms into pairs or sets that have a common factor.
Here’s how it’s done for the polynomial given in the exercise:
First, rearrange and group the terms: ewline ax + ay + 3x + 3y = (ax + ay) + (3x + 3y)
Next, look inside each group to identify common factors.
Factor out the common factor from each group:
(ax + ay) + (3x + 3y) = a(x + y) + 3(x + y)
You will notice that both grouped terms now have a common binomial factor, (x + y).
The next step involves factoring out this common binomial, making factoring by grouping a powerful tool for simplifying polynomials:
a(x + y) + 3(x + y) = (x + y)(a + 3)

The Greatest Common Factor (GCF) is the largest number or term that divides all terms in an expression.
Identifying and factoring out the GCF simplifies the polynomial.
In our polynomial \((a x + a y + 3 x + 3 y)\), we look at grouped terms \((ax + ay)\) and \((3x + 3y)\).
For each group, identify the common factor:
- In \((ax + ay)\), the GCF is 'a' since it is common to both terms.
- In \((3x + 3y)\), the GCF is '3'.
So, we factor 'a' from the first group and '3' from the second:
\((a(x + y) + 3(x + y))\).
Now notice, \((x + y)\) is a common binomial factor in both terms. This makes it easier to perform additional factorization.

Identifying the GCF helps to significantly reduce the polynomial, aiding in easier manipulation and simplification.

A binomial is an algebraic expression with exactly two terms.
Factoring out a common binomial is an essential step in simplifying polynomials.
In our example of the polynomial \(a x + a y + 3 x + 3 y\), after grouping and factoring out the GCF from each group, we have:
\(a(x + y) + 3(x + y)\).
Here, \((x + y)\) is the common binomial factor.
Factoring it out, you get:
\((x + y)(a + 3)\).

This process of identifying the common binomial factor and factoring it out helps to break down and simplify even the most complex polynomial expressions.
It not only reduces the polynomial but also paves the way for solving polynomial equations more effectively.