When you see an expression like $(a+b)²$, it's tempting to just square each term inside, getting $a² + b²$. But that's a common mistake. The exponent applies to the entire binomial, meaning we're multiplying $(a+b)$ by itself.

To multiply two binomials, we can use a handy mnemonic called the FOIL method—it ensures we multiply every term in the first binomial by every term in the second. FOIL stands for First, Outer, Inner, Last.

Applying this to $(a+b)(a+b)$ gives us four terms:

• First: $a \cdot a = a²$
• Outer: $a \cdot b = ab$
• Inner: $b \cdot a = ba$
• Last: $b \cdot b = b²$

Putting them all together, we get $a² + ab + ba + b²$.

Notice the two middle terms: $ab$ and $ba$. Because of the commutative property of multiplication, the order doesn't matter, so $ab$ is the same as $ba$. This means we can combine these like terms.

By simplifying, we arrive at a general formula. This is the binomial square identity, a reliable shortcut that saves you from having to FOIL every time.

The same logic applies to squaring a binomial with subtraction, like $(a-b)²$. We can think of this as $(a + (-b))²$ or just apply FOIL directly to $(a-b)(a-b)$.

• First: $a \cdot a = a²$
• Outer: $a \cdot (-b) = -ab$
• Inner: $(-b) \cdot a = -ba$
• Last: $(-b) \cdot (-b) = b²$

Combining these gives us $a² - ab - ba + b²$. Again, we combine the two like middle terms.

This leads to the second key binomial square identity.

Memorizing these two identities will significantly speed up your work in algebra. Let's test your understanding.

Mastering these patterns is a foundational step for factoring, solving quadratic equations, and understanding more complex polynomials.