Polynomials (many terms) are algebraic expressions formed by adding or subtracting monomials (single terms with positive exponents or constants). A Binomial is a polynomial containing two terms, bi meaning two.

In terms of nomenclature, 'bi' means 2 hence binomial means two terms. So in general, a binomial is a polynomial containing two terms. The general form can be written as axn ± bym
A different and simpler way of defining a binomial is that it is an algebraic expression containing two terms connected by a sum of a difference sign. Example: 3x+5y,a+3x, x2-3x…etc.

An algebraic expression which contains two terms is called a binomial.

Examples:

2x -3 This binomial is formed by subtracting 3 from 2x.
x2 + 4y2 The term 4y2 is added to x2 to give a Binomial expression.
The product of many variables yield only monomials
3xyz has three variables namely x, y and z but it is a single term or a monomial. The constant 3 is the coefficient of the term xyz.

More examples for Binomials:
5abc + 6xy
ax + by
$\frac{x^3}{3}$+ $\frac{y^2}{2}$

The verbal expressions are often expressed as Binomials.

Verbal Expressions Algebraic Expression
Three more than a number x + 3
Five added to square of a number z2 + 5
The length of a rectangle is 4 less
than twice the width
2w - 4
Dan's age is ten less than half his father's age $\frac{x}{2}$ - 10
sum of the squares on the legs of a right triangle a2 + b2

When two binomials are added, the like terms are combined together. So in such cases, addition can be done between those terms that have the same variable with the same exponent.
Example:
(3a+5y)+ (a+4y) = (3a+a) + (5y+4y)
=4a+9y

How ever 5y+y2 cannot be added so we write it as it is 5y+y2. This is because its not possible to add unlike terms.
When two binomials are being added, try to gather the like terms. By doing this the like terms come together and simplification is made simple. In case there are any like terms missing we can complete it by using 0.

For example:
(x+2y)+y= (x+2y) + (0x+y)
=(x+0x) + (2y+y)
= (1+0) x+ (2+1) y
=x+3y

When two binomials are subtracted, the like terms are combined together.

Example:
(x+5y)- (2x+7y) =(x-2x) + (5y-7y)
=-x+ (-2y)
=-x-2y

This operation can also be done in another way that is reversing the sign and then the terms can be added.
(x+5y)- (2x+7y) =(x+5y) + (-2x-7y) -(x-2x) + (5y-7y)
=-x+ (-2y)
=-x-2y

The product of two binomial forms a trinomial.

Product of two binomials can be found using a method called “FOIL”.
F-front
O- outer
I-inner
L-last

Consider for example:
(2x+y) $\times$ (6y+x)
F-here the front terms of both the binomials are taken and multiplication is performed.
Here front terms are 2x and 6y.
Multiplying them=2x $\times$ 6y=12xy
O-here both the outer terms of the binomials are taken and multiplied.
Here outer terms are 2x and x.
Multiplying them=2x $\times$ x=2x2
I-here both inner terms are taken and multiplied.
Here inner terms are y and 6y.
Multiplying them=y $\times$ 6y=6y2
L-here the last terms of the binomials are taken and multiplied.
Here last terms are y and x.
Multiplying them=y $\times$ x=xy

So (2x+y) $\times$ (6y+x) =12xy+2x2+6y2+xy
Now like terms are combined so = 13xy+2x2+6y2

Example:
(y + 3)(y+4) = (y +…) (y +…) = y2
= (y +…) (…+ 4) = 4y
= (…+ 3)(x +…) = 3y
= (…+ 3) (…+ 4) = 12

(y + 3)(y + 4) = y2+4y+3y+12
= y2+7y+12

When dealing with the negative terms in a binomial each step must be taken carefully.
Two methods can be employed for doing this.

Method I
Example: (x+5)(3x-2)
Here, first of all insert the parenthesis “[ ]”
So – [(x+5)(3x-2)]
Then use FOIL method
– [3x2-2x+15x-10]
Simplify
- [3x2+13x-10]
Remove [ ] and change the sign
-3x2-13x+10
So, -(x+5) (3x-2) = -3x2-13x+10

Method II

Example: -(x+5) (3x-2)
First distribute the sign for first binomial
(-x-5)(3x-2)
Expand using FOIL method
-3x2+2x-15x+10
Simplify
-3x2-13x+10

When binomials are divided it is first important to arrange the divisor and the dividend in order (ascending or descending order) of some variable used in it.
There are mainly two ways of dividing:

Method I:
Take the dividend’s first term and divide it by the divisor’s first term and write the value as the quotient. In the next step, take the quotient obtained and multiply the divisor by it and place the respective value below the dividend keeping the like terms under each other. Subtract this from the dividend, take the reminder and any remaining terms from the dividend as the new dividend and the process once again.

Example:
$\frac{(x^{2} - 4)}{(x - 2)}$
Here dividend =x2-4
Divisor = x-2
The following process is performed
(x-2) ) x2+0x-4 ( x+2
x2-2x
(-)_______
2x – 4
2x – 4
(-)_______
0
Answer = x+2

Method II:
The binomial can be factored using the algebraic formulas mentioned above:
Then the common factors can be cancelled.

Example:
$\frac{(x^{2} - 4)}{(x - 2)}$ = $\frac{(x^{2} - 2^{2})}{(x - 2)}$ = $\frac{((x - 2) (x + 2))}{(x - 2)}$ =(x+2)
  • Factoring can be done by taking the GCF of the two terms of the binomial and factoring that out.
For example:
4x - 6x2
Here the GCF= 2x
So 4x-6x2 = 2x (2-3x)
  • Another way is to use special algebraic formulas:
So general used form are
$\Rightarrow$ Sum and difference
(x+y)(x-y)=x2 - y2

$\Rightarrow$ Binomial squared
(x+y)2 = x2 + 2xy + y2

$\Rightarrow$ Binomial cube
(x+y)3 =x3+3x3y+3xy3+y3