Proof of the Square of a Binomial: (ab)² = a²  2ab + b²
This proof demonstrates the expansion of the square of a binomial (ab), which is frequently used in algebra and other mathematical fields.
Understanding the Concept
The square of a binomial (ab)² means multiplying the binomial by itself:
(a  b)² = (a  b)(a  b)
Using the Distributive Property
We can expand this product using the distributive property:

Distribute the first term (a) of the first binomial to the second binomial: a(a  b) = a²  ab

Distribute the second term (b) of the first binomial to the second binomial: b(a  b) = ab + b²

Combine the results from steps 1 and 2: a²  ab  ab + b²
Simplifying the Expression
Finally, combine the like terms:
a²  2ab + b²
Therefore, we have proven that:
(a  b)² = a²  2ab + b²
Visualizing the Proof
We can also visualize this proof using a geometric representation:
Imagine a square with sides of length (a  b). The area of this square is (a  b)². We can divide this square into four smaller regions:
 A square with side length 'a' (area a²)
 A rectangle with sides of length 'a' and 'b' (area ab)
 Another rectangle with sides of length 'a' and 'b' (area ab)
 A square with side length 'b' (area b²)
The total area of the large square is equal to the sum of the areas of the four smaller regions:
(a  b)² = a²  ab  ab + b² = a²  2ab + b²
This visual representation further clarifies the proof and reinforces the concept.