Expanding Binomials: Understanding the Distributive Property
The equation (x + a)(x + b) = (x + a)x + (x + a)b demonstrates the distributive property in action, a fundamental concept in algebra. This article explores how the distributive property helps us expand binomials and arrive at a simplified expression.
The Distributive Property in Action
The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the results. In our case:

(x + a) is multiplied by (x + b). This means we need to distribute (x + a) to both terms inside the second set of parentheses.

First, we multiply (x + a) by x, resulting in (x + a)x.

Next, we multiply (x + a) by b, resulting in (x + a)b.
Therefore, the equation (x + a)(x + b) = (x + a)x + (x + a)b accurately reflects the application of the distributive property.
Expanding and Simplifying
The equation (x + a)x + (x + a)b can be further simplified by applying the distributive property again:

(x + a)x = x² + ax

(x + a)b = bx + ab
Combining these results, we get:
(x + a)(x + b) = x² + ax + bx + ab
This expanded form is now a trinomial (an expression with three terms). It represents the simplified version of the original product of binomials.
Importance of the Distributive Property
Understanding the distributive property is crucial for several reasons:

Expanding and simplifying algebraic expressions: It allows us to express complex expressions in a simpler, more manageable form.

Solving equations: The distributive property helps us isolate variables and solve equations.

Factoring: It is also the basis for factoring quadratic equations, which is essential for solving them.
In conclusion, the distributive property plays a vital role in algebra, enabling us to expand, simplify, and solve various mathematical expressions. By understanding this fundamental concept, we can manipulate algebraic expressions effectively and confidently.