91Ó°ÊÓ

Factoring polynomials is a powerful tool in algebra. It involves breaking down a polynomial into simpler 'factors' that, when multiplied together, give you the original polynomial. Think of it like unboxing a package into smaller, more manageable pieces. A polynomial is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s). For example, the polynomial given in the exercise is:
y^{2} - 6y + yw - 6w.
By factoring, we make complex expressions simpler and easier to solve or analyze.

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers. In algebra, it can also be the largest expression that divides terms of a polynomial. To find the GCF of a set of algebraic terms, you typically look for common variable factors and constants. Let's recall the exercise:
y^{2} - 6y + yw - 6w.
We grouped the terms:

• (y^{2} - 6y) and
• (yw - 6w)

Now, we factor out the GCF from each group:
- For the first group (y^{2} - 6y), the GCF is 'y'. Factoring out 'y', we get: y(y - 6).
- For the second group (yw - 6w), the GCF is 'w'. Factoring out 'w', we get: w(y - 6).
So this step helps in breaking things down to smaller units that can be further simplified.

A binomial is an algebraic expression containing exactly two terms. In the given exercise, notice that we have two binomials after factoring out the GCF from each pair. These binomials are:
y(y - 6) + w(y - 6).
Next, we look for common binomials, the structures that are the same in the factored terms. In our case, (y - 6) appears in both terms. This is crucial as it allows further factoring. Factoring out the common binomial factor (y - 6), we get:
(y - 6)(y + w).
So our original polynomial y^{2} - 6y + yw - 6w is now factored into a product of two binomials: (y - 6)(y + w).
Using these steps repeatedly will make polynomial factoring much simpler and easier to understand. Happy learning!