91Ó°ÊÓ

Understanding the Greatest Common Factor (GCF) is crucial to factoring polynomials. The GCF of two or more terms is the largest factor that all terms can be divided by evenly. In the polynomial \(xy - 3y + x - 3\), each term has individual factors.
For example:

• In \(xy\), the factors are \(x\) and \(y\).
• In \(3y\), the factors are \(3\) and \(y\).
• In \(x\), the factor is \(x\).
• In \(3\), the factor is \(3\).

When we group \(xy - 3y\) and \(x - 3\), finding the GCF of each binomial becomes clear. For \(xy - 3y\), \(y\) is the GCF. For \(x - 3\), there is no common factor other than 1.

Polynomial factorization involves breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. This can be thought of as the reverse of expanding a polynomial. Factoring helps in simplifying expressions and solving polynomial equations.
In the given example, the polynomial is \(xy - 3y + x - 3\). Our goal is to rewrite this as a product of simpler polynomials. By grouping the terms \((xy - 3y) + (x - 3)\)and factoring out the GCF from each group, we get closer to this goal. This step ensures that we transform the polynomial into a form where common factors can be spotted easily.

Factoring by grouping is a method used when a polynomial contains four or more terms. The idea is to group terms with common factors and then factor out these common factors. Here’s a step-by-step breakdown:

1. **Group Terms:** Arrange and group the polynomial into pairs: \((xy - 3y) + (x - 3)\).
2. **Factor Each Group:** For each group, factor out the GCF. For \(xy - 3y\), the GCF is \(y\): \(y(x - 3)\). For \(x - 3\), it stays as \(1(x - 3)\) because 1 is the only common factor.
3. **Factor out the Common Binomial:** Notice \((x - 3)\) appears in both groups. It becomes the common binomial factor. So, combine this into: \((x - 3)(y + 1)\). This is the factored form of the original polynomial.
Understanding this method allows you to break down more complex polynomials and solve equations much more easily.