Binomial

A binomial is an algebraic expression consisting of the sum or difference of two terms: $$ a + b \quad \text{or} \quad a - b $$ where \( a \) and \( b \) are the terms of the binomial. The sign between the terms can be either a plus \(+\) or a minus \(-\).

In other words, a binomial is a specific type of polynomial made up of exactly two monomials.

Each term can be a constant, a variable, or a product involving both numbers and variables.

Practical Example

For example, a binomial can be formed by adding a variable and a constant:

$$ x + 3 $$

It can also be formed by subtracting a constant from a term with a variable:

$$ 2y - 5 $$

Another example is the sum of a squared term and a variable:

$$ 3a^2 + b $$

Working with Binomials

Binomials follow the standard rules of algebra. The most common operations include:

1] Addition and Subtraction of Binomials

To add or subtract binomials, combine the like terms.

For example:

$$ (x + 3) + (2x - 5) = (x + 2x) + (3 - 5) = 3x - 2 $$

2] Multiplication of Binomials

A well-known case is the product of two binomials, which can be expanded using the distributive property:

$$ (a + b)(c + d) = ac + ad + bc + bd $$

Here’s a concrete example:

$$ (x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6 $$

3] Squaring a Binomial

The square of a binomial is a classic special product:

$$ (a + b)^2 = a^2 + 2ab + b^2 $$

This identity comes from multiplying the binomial by itself:

$$ (a + b)(a + b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2 $$

For example:

$$ (x + 3)^2 = x^2 + 6x + 9 $$

And so on.