A binomial is an algebraic expression consisting of two different terms, represented by \(a\) and \(b\), combined with an addition or subtraction operator. Specifically, a binomial can be written as:

\[ (a + b)\ \quad \text{or} \quad (a-b) \]

The terms \(a\) and \(b\) belong to a specific algebraic space, such as real numbers, complex numbers, or elements of an arbitrary field. These terms can include variables, constants, or products of variables and constants.

Binomials possess some properties that make them useful for manipulation. Among these are the notable or remarkable products. Notable products are specific products of powers, binomials, and trinomials. These products are essential for solving equations and also serve as a key to unlocking common mathematical patterns. For a more comprehensive understanding, please refer to the topic page.


Associative property

The associative property can be applied to binomials, stating that grouping terms in a sum or product of binomials does not affect the final result. This means that if we have three binomials, such as \((a + b)\), \((c + d)\), and \(e + f)\), the associative property allows us to regroup the terms within a sum or product without changing the overall value. For instance, if we add these binomials in the order \((a + b) + (c + d) + (e + f)\), we get:

\begin{align} (a + b) + (c + d) + (e + f) &= a + b + c + d + e + f\\[0.6em] &= (a + (b + c)) + ((d + e) + f) \end{align}

Similarly, for the multiplication of binomials, if we have three binomials \((a + b)\), \((c + d)\), and \(e + f)\), the associative property allows us to regroup the factors within a product without affecting the overall product:

\begin{align} (a + b) \cdot (c + d) \cdot (e + f) &= a \cdot b \cdot c \cdot d \cdot e \cdot f \\[0.6em] &= (a \cdot (b \cdot c)) \cdot ((d \cdot e) \cdot f) \end{align}


Distributive property

The distributive property is a fundamental mathematical concept that describes the relationship between multiplication and addition or subtraction. It states that the product of a term and the sum or difference of two other terms equals the sum or difference of the products of the first term with each term of the second expression. This property is widely used in algebra and forms the basis for many operations. In other words:

\[ a(b + c) = ab + ac \]

\[ a(b – c) = ab – ac \]

where \(a\), \(b\), and \(c\) are any real numbers or algebraic expressions.


Commutative property

The commutative property allows for manipulating the order of terms and factors in binomials, resulting in a simplified and efficient method of working with them. It states that the order of terms in the addition or multiplication of binomials does not affect the final result. Specifically, when two binomials, \((a + b)\) and \((c + d)\), are considered, the commutative property allows for rearranging terms within a sum or product without altering the overall value.

For example, if \((a + b)\) and \((c + d)\) are added in the order \((a + b) + (c + d)\), the result will be \((a + c) + (b + d)\) after the commutative property is applied.

\[(a + b) + (c + d) = (a+c) + (b+d)\]

Similarly, for the multiplication of binomials, the commutative property allows for rearranging factors without affecting the overall product. For example, if \((a + b)\) and \((c + d)\) are multiplied in the order \((a + b) \cdot (c + d)\), the result will be \((a \cdot c) + (a \cdot d) + (b \cdot c) + (b \cdot d)\) after the commutative property is applied.