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When multiplying binomials, one essential concept is the *distributive property*. This property states that you can distribute each term in the first binomial to every term in the second binomial. If we break this down:
Imagine you have two binomials, like \(a + b\) and \(c + d\). Using the *distributive property*, you get:
\[ (a + b)(c + d) = a(c + d) + b(c + d) \]
Each term in the first binomial (\(a\) and \(b\)) gets multiplied by each term in the second binomial (\(c\) and \(d\)). This ensures every possible product between the terms is accounted for. When multiplying \( (y - 6)(y - 2) \), the distributive property is applied. Each term in \(y - 6\) is distributed over \(y - 2\):

• \(y \cdot y \)


• \(y \cdot (-2)\)


• \((-6) \cdot y \)


• \((-6) \cdot (-2)\)

Following this property makes binomial multiplication more structured and easier to manage.

A more specific application of the distributive property for multiplying binomials is the *FOIL method*. FOIL stands for First, Outside, Inside, Last. This method ensures that all the products of the terms within the binomials are covered.

Let's revisit multiplying \(y - 6\) and \(y - 2\) using FOIL:

First: Multiply the first terms in each binomial:
\[ y \cdot y = y^2 \]

Outside: Multiply the outer terms in the product:
\[ y \cdot (-2) = -2y \]

Inside: Multiply the inner terms:
\[ (-6) \cdot y = -6y \]

Last: Multiply the last terms in each binomial:
\[ (-6) \cdot (-2) = 12 \]

The FOIL method helps make sure no product is missed. Once you've found all the products, you can move to the next step: combining like terms.

After using the distributive property or the FOIL method, you often get a polynomial with several terms. The next important step is *combining like terms*. Like terms are terms that contain the same variable raised to the same power.

Consider the results from FOIL applied to \( (y - 6)(y - 2) \):\[ y^2 - 2y - 6y + 12 \]

Notice the terms \(-2y\) and \(-6y\)? These are like terms because they both have \(y\) as their variable. Combining them means summing their coefficients:
\[ -2y - 6y = -8y \]
Now, we rewrite the expression, combining the like terms:

\[ y^2 - 8y + 12 \]

This simplifies the polynomial and gives us our final product. Combining like terms is essential as it reduces the expression to its simplest form, making it easier to interpret and use in further calculations.