GRE Algebra | Operations with Algebraic Expressions

An expression contain variables, numbers and operation symbols is called an algebraic expression.Every expression can be written as a single term or sum of terms.
Here are some examples of algebraic expressions.

(1): 5x + 2y + 7 
(2): 2x2 + 3y2 + 5
(3): x3z + 2x2 + 3x + 9
(4): 4x/(2x + 1) 

A number multiplied by the variables is called coefficient of a term.
In above example (2), 2 is coefficient of 2x² and 3 is the coefficient of 3y² and 5 is the constant.

Operations performed on algebraic expression are:

1. Addition & Subtraction:
   On performing addition or subtraction on algebraic expression the coefficients of same degree added or subtracted.
   For example:

   => 3x + 4x = 7x
   => a3 + 4a2 - 3a2 + 2 = a3 + a2 + 2 

2. Multiplication:
   Two algebraic expressions can be multiplied by multiplying each term of first expression to the each term of the second expression.
   For example:

   => (3a + 3)(2a - 8) = 3a(2a) + 3a(-8) + 3(2a) - 3(8)
                       = 6a2 - 24a + 6a - 24
                       = 6a2 - 18a - 24  

3. Common Factor:
   A number or variable can be factored out of each term of expression if it is common in all the terms.
   For example:

   => 3y + 15 = 3(y + 5)
   => 9x2 - 3x = 3x(3x - 1)
   => 4y2 + 8y/ 2y+ 4 = 4y(y + 2)/ 2(y+2) =  4y/2 (where y ≠ 2 ) 

4. Identity:
   It can be defined as a statement of equality between two algebraic expression and it is true for all possible values.
   For example:

   (x + y)2 = x2 + 2xy + y2
   (x - y)2 = x2 - 2xy + y2
   (x + y)3 = x3 + 3x2y + 3xy2 + y3
   (x - y)3 = x3 - 3x2y + 3xy2 - y3
   x2 - y2 = (x + y)(x - y)
   x3 - y3 = (x - y)(x2 + xy + y2) 
   x3 + y3 = (x + y)(x2 - xy + y2) 
   x2 + y2 + z2 = (x + y + z)2 - 2(xy + yz + zx)
   x3 + y3 + z3 - 3xyz = (x + y + z)(x2 + y2 + z2 - xy - yz - zx)
   (x + y)(y + z)(z + x) = (x + y + z)(xy + yz + zx) - xyz 

An equation is true only for certain values of variables.

1. Linear equation in one variable:

   2a + 4 = 8 

2. Linear equation in two variables:

   5a + 7b = 49 

3. A quadratic equation in one variable:

   4a2 + 2a = 16