# 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:

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 `

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