# DEFINITION OF BINOMIAL

Binomial is an algebraic expression which contains exactly two terms either in addition or subtraction.

Examples :

x + y

2a - b

p + 4q

x2 + 3xy

p + 8

What if two terms are in multiplication or division?

An algebraic expression in which two terms are in multiplication or division is not binomial.

For example, x2y is not a binomial. Even though we have two terms x2 and y in x2y,  we can not consider x2y as binomial. Because x2 and y are in multiplication.

## Algebraic Identities of Binomials

(a + b)2 = a2 + 2ab + b2

(a - b)2 = a2 - 2ab + b2

a2 - b2 = (a + b)(a - b)

(a + b)3 = a3 + 3a2b + 3ab2 + b3

or

= a3 + 3ab(a + b) + b3

(a - b)= a3 - 3a2b + 3ab2 - b3

or

= a3 - 3ab(a - b) - b3

a3 + b= (a + b)3 - 3ab(a + b)2

a3 - b= (a - b)3 + 3ab(a - b)2

To add two or more binomials, we have to group the like terms and combine them.

Example 1 :

Simplify :

(x + y) + (2x - 3y)

Solution :

= (x + y) + (2x - 3y)

= x + y + 2x - 3y

Group the like terms and combine them.

= (x + 2x) + (y - 3y)

= 3x + (-2y)

= 3x - 2y

Example 2 :

Simplify :

(5x2y + 3) + (5 - 3x2y)

Solution :

= (5x2y + 3) + (5 - 3x2y)

= 5x2y + 3 + 5 - 3x2y

Group the like terms and combine them.

= (5x2y - 3x2y) + (3 + 5)

= 2x2y + 8

Example 3 :

Simplify :

(3xy + 3) + (5x + xy)

Solution :

= (3xy + 3) + (5x + xy)

= 3xy + 3 + 5x + xy

Group the like terms and combine them.

= (3xy + xy) + 5x + 3

= 4xy + 5x + 3

Example 4 :

Simplify :

(x2 + 3xy2) + (5x2 - 3x2y) + (7xy2 - 5x2y)

Solution :

= (x2 + 3xy2) + (5x2 - 3x2y) + (7xy2 - 5x2y)

= x2 + 3xy2 + 5x2 - 3x2y + 7xy2 - 5x2y

Group the like terms and combine them.

= (x2 + 5x2) + (-3x2y - 5x2y) + (3xy+ 7xy2)

= 6x2 + (-8x2y) + 10xy2

= 6x2 - 8x2y + 10xy2

## Subtracting Binomials

To subtract two binomials, distribute the negative sign to the terms in the second binomial and group the like terms and combine them.

Example 5 :

Simplify :

(a + 2b) - (2a + b)

Solution :

= (a + 2b) - (2a + b)

Distribute the negative sign.

= a + 2b - 2a - b

Combine the like terms by grouping.

= (a - 2a) + (2b - b)

= -a + b

= b - a

Example 6 :

Simplify :

(7pq + 4) - (3p - 5pq)

Solution :

= (7pq + 4) - (3p - 5pq)

Distribute the negative sign.

= 7pq + 4 - 3p + 5pq

Combine the like terms by grouping.

= (7pq + 5pq) - 3p + 4

= 12pq - 3p + 4

Example 7 :

Simplify :

(7xy2 + 5) - (3 + 2xy2)

Solution :

= (7xy2 + 5) - (3 + 2xy2)

Distribute the negative sign.

= 7xy2 + 5 - 3 - 2xy2

Combine the like terms by grouping.

= (7xy2 - 2xy2) + (5 - 3)

= 5xy2 + 2

Example 8 :

Simplify :

(3x2 - 5x2y) - (2x2y - 7x2)

Solution :

= (3x2 - 5x2y) - (2x2y - 7x2)

Distribute the negative sign.

= 3x2 - 5x2y - 2x2y + 7x2

Combine the like terms by grouping.

= (3x2 + 7x2) + (-5x2y - 2x2y)

= 10x2 + (-7x2y)

= 10x2 - 7x2y

## Multiplying Binomials

You can use the following methods to multiply a binomial by a binomial.

(i) Distributive Property.

(ii) FOIL Method

## Distributive Property

To multiply a binomial by a binomial, Distributive Property can be used more than once. ## FOIL Method

Another method for multiplying binomials is called the FOIL method.

1. Multiply the First terms. 2. Multiply the Outer terms. 3. Multiply the Inner terms. 4. Multiply the Last terms.  Example 9 :

Multiply using Distributive Property.

(p + 2)(p - 5)

Solution :

= (p + 2)(p - 5)

Distribute.

= p(p - 5) + 2(p - 5)

Distribute again.

= p2 - 5p + 2p - 10

Combine the like terms.

= p2 - 3p - 10

Example 10 :

Expand (x + y)using Distributive Property.

Solution :

(x + y)2

Write as a product of two binomials.

(x + y)(x + y)

Distribute.

= x(x + y) + x(x + y)

Distribute again.

= x2 + xy + xy + y2

Combine the like terms.

=  x2 + 2xy + y2

Example 11 :

Multiply using FOIL method.

(5x2 - 2y)(x2 + 3y)

Solution :

= (5x2 - 2y)(x2 + 3y)

Use the FOIL method.

= (5x⋅ x2) + (5x2⋅ 3y) + (-2y ⋅ x2) + (-2y ⋅ 3y)

Multiply.

= 5x4 + 15x2y + (-2x2y) + (-6y2)

= 5x4 + 15x2y - 2x2y - 6y2

Combine the like terms.

= 5x4 + 13x2y - 6y2

Example 12 :

Expand (y - 3)using FOIL method.

Solution :

= (y - 3)2

Write as a product of two binomials.

= (y - 3)(y - 3)

Use the FOIL method.

= (y ⋅ y) + (y ⋅ -3) + (-3 ⋅ y) + (-3 ⋅ -3)

Multiply.

= y2 + (-3y) + (-3y) + 9

= y2 - 3y - 3y + 9

Combine like terms.

= y2 - 6y + 9

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