91Ó°ÊÓ

To understand the 'difference of squares' pattern, we need to first look at binomial expansion. A binomial is a polynomial with two terms, like \(a - b\) or \(a + b\). Expanding these binomials means multiplying them together to form a new expression.

When we talk about expanding \(a - b\)(a + b)\, we are interested in finding the result of their multiplication. This begins by using the distributive property.

Notice that each term in each binomial must be multiplied by each term in the other binomial. This can seem complicated at first, but with practice, it becomes easier.

First, we do \(a \cdot a\), which gives us \(a^2\). Next, we do \(a \cdot b\), which gives us \(ab\). Then, we do \(-b \cdot a\), which gives us \(-ab\). Finally, we do \(-b \cdot b\), which gives us \(-b^2\).

Putting it all together, we have: \(a^2 + ab - ab - b^2\).

The distributive property is a crucial concept in algebra and helps us expand binomials effectively. This property states that when you multiply a number by a sum, you can distribute the multiplication over each addend.

In mathematical terms, it is expressed as \(a(b + c) = ab + ac\).

When applying this to the binomial expansion \(a - b \cdot a + b\), we distribute each term in the first binomial to each term in the second binomial, one by one.

Here's a step-by-step of how we apply the distributive property:

• First, we multiply \(a\) by both \(a\) and \(b\): \(a \cdot a = a^2\) and \(a \cdot b = ab\).
• Then, we multiply \(-b\) by both \(a\) and \(b\): \(-b \cdot a = -ab\) and \(-b \cdot b = -b^2\).

This gives us: \(a^2 + ab - ab - b^2\). In this process, combining like terms becomes necessary, which we'll discuss next.

Understanding like terms is important to simplify expressions effectively. Like terms are terms that have the same variables raised to the same powers. For example, \(3a\) and \(-5a\) are like terms because they both involve the variable \(a\).

In our expanded expression from the binomial expansion \(a - b \cdot a + b\), we end up with \((a^2 + ab - ab - b^2)\).

Notice the \(ab\) and \(-ab\) terms. These are like terms because they both have the variable \(ab\). When we combine these, they cancel out each other, creating a simpler expression:

• \