# Algebraic Expressions and Identities

## What is algebraic expression

Algebraic expression is the expression having constants and variable. It can have multiple variable and multiple power of the variable
Example
11x
2y – 3
2x + y
2xy + 11z=1
x2 +2+x

## Terms, Factors and Coefficients

Let’s explain this with an example
Example
11x + 1
Here are the Important points
1) This expression is made up of two terms, 11x and 1.
2) Terms are added to form algebraic expressions.
3) Terms themselves can be formed as the product of factors. The term 11x is the product of its factors 11 and x. The term 1 is made up of just one factor, i.e., 1.
4) The numerical factor of a term is called its numerical coefficient or simply coefficient. The coefficient in the term 11x is 11 and the coefficient in the constant term  1 is 1.

## Monomials, Binomials, trinomial and Polynomials

### Monomials

Algebraic expression having one terms is called monomials
Example
11x2
6xy
–2z
10y
–9
82mnq

### Binomials

Algebraic expression having two terms is called Binomials
Example
x + xy
2p + 11q
y + 1
11 –3xy

### Trinomials

Algebraic expression having three terms is called Trinomials
Example
x + xy+1
2p + 11q+pq
y + 2+y2
11 –3xy+x

### Polynomials

In general, an expression containing, one or more terms with non-zero coefficient (with variables having non negative exponents) is called a polynomial
Example
x + y + z + 1
3pq
11xyz – 10x
4x + 2y + z

Watch this tutorial for more explanation about polynomials,monomials,binomials

## Like and Unlike Terms

It is quite important concept and quite well used in addition, subtraction and multiplication of the algebraic expression.
When the variable part of the terms is same, they are called like terms
Example
2x, 3x are like term
5x2 and –9x2  are  like terms.
Like term can be add or subtracted. Basically We need to add or  subtract the coefficent
When the variable part of the terms is not same, they are called unlike terms
Example
2x, 3y are unlike term
5x2 and –9zx2  are  unlike terms.

## Addition and Subtraction of Algebraic Expressions

We many times need to add the two algebraic expression. Adding algebraic expression just means adding the like terms. We need to follow below steps for Addition of algebraic expression
1) We write each expression to be added in a separate row. While doing so we write like terms one below the other
Or
We add the expression together on the same line and arrange the like term together
3) Write the Final algebraic expression
Example
Add the following expression x – y + xy, y – z + yz, z – x + zx
Solution
= (x – y + xy)+( y – z + yz)+( z – x + zx)
Arranging the like term together
= x-x -y+y-z+z+xy+yz+zx
=xy+yz+zx   as x-x=y-y=z-z=0

### Subtraction

Similarly, for subtracting algebraic expression
1) We write each expression to be subtracted in a separate row. While doing so we write like terms one below the other   and then we change the sign of the expression which is to be subtracted i.e. + becomes – and – becomes +
Or
We subtract the expression together on the same line, change the sign of all the term which is to be subtracted and then arrange the like term together
3) Write the Final algebraic expression
Example
Subtract 4x – 7xy + 3y + 12 from 12x – 9xy + 5y – 3
Solution
=(12x – 9xy + 5y – 3) – (4x – 7xy + 3y + 12)
While subtracting, we need to remember signs are reversed after –sign once bracket is opened
ie. + becomes - and - becomes +
=12x-9xy+5y-3 -4x+7xy-3y-12
Arranging the like term together
=8x-2xy+2y-15

Watch this tutorial for questions on addition and subtraction

## Multiplication of Algebraic expression

General steps for Multiplication
1) We have to use distributive law and distribute each term of the first polynomial to every term of the second polynomial.
2) when you multiply two terms together you must multiply the coefficient (numbers) and add the exponents
3) Also as we already know ++ equals =, +- or -+ equals - and -- equals +
4) group like terms
Multiplication may involve
a) Multiplication of monomial to monomial
b) Multiplication of monomial to binomial, trinomial or more terms polynomials
c) Multiplication of binomial, trinomial or more terms polynomials to monomial
d) Multiplication of binomial to binomial, trinomial or more terms polynomials
e) Multiplication of trinomial to trinomial or more terms polynomials

Multiple the Monomials
1. a2 × (2b22) × (4a26)
2 (-10pq3/3) × (6p3/5)

Answer: We will use the below property extensively in above questions
xm × xn × xo = xm+n+o
1. As you know
So, we get
a2 × (2b22) × (4a26) = 8a28b22
2. (10pq3/3) × (6p3/5)
=(-12p4q3)

Multiply the binomials.
(i) (2a + 6b) and (4a – 3b)
(ii) (x – 1) and (3x – 2)
Answer: Let  ( a+b) (c+d) to be done
then by distributive law
( a+b) (c+d)= a(c+d) + b( c+d)
=(a ×  c)+(a ×  d)+(b ×  c)+(b × d)
We will use the same concept in all  the question below
1) (2a + 6b) and (4a – 3b)
= 2a × 4a – 2a × 3b + 6 b × 4a – 6b × 3b
= 8a² - 6ab + 24ab -18b2
= 8a² + 18ab -18b2
2) (x – 1) and (3x – 2)
= x × 3x – 2x - 1 × 3x + 2
= 3x2 -2x – 3x + 2
= 3x2 – 5x + 2

Multiply Binomial by Trinomial
(3x + 1)(4x2 – 7x + 1)
By distributive law
=3x(4x2 – 7x + 1) + 1(4x2 – 7x + 1)
=12x3 -21x2 +3x +4x2 -7x+1
=12x3 -17x2 -4x+1
Multiply Trinomial by Trinomial
(a + b + c)(a + b c)
By distributive law
=a(a + b –c) + b(a + b –c)+c(a + b –c)
=a2 +ab-ac +ab +b2 -bc +ac+bc-c2
=a2+ b2 -c2 +2ab
Watch this tutorial for questions on multiplication of monomials with monomials

Watch this tutorial for questions on multiplication of binomails with binomials

## What is an Identity

An identity is an equality, which is true for all values of the variables in the equality.
(a + b)2 = a2 + 2ab + b2
It is true for all the values of a and b
On the other hand, an equation is true only for certain values of its variables. An equation is not an identity
x2    =1
The below four identities are useful in carrying out squares and products of algebraic expressions.
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
(a + b) (a b) = a2b2
(x + a) (x + b) = x2 + (a + b) x + ab
These allow easy alternative methods to calculate products of numbers.
Example
Use a suitable identity to get each of the following products.
(i) (y + 1) (y + 1)
(ii) (2x + 1) (2x -1)
(iii) (2z – 3) (2z – 3)
Solution
i) (y + 1) (y + 1)
=(y+1)2
Now (a + b)2 = a2 + 2ab + b2
=y2 +2y+1
ii) (2x + 1) (2x -1)
Now (a + b) (a b) = a2b2
=4x2 -1
iii) (2z – 3) (2z – 3)
=(2z-3)2
Now as (a - b)2 = a2 - 2ab + b2
=4z2 -12z+9
Watch this tutorial for questions on identities