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):2x^{2}+ 3y^{2}+ 5(3):x^{3}z + 2x^{2}+ 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 2x^{2} and 3 is the coefficient of 3y^{2} and 5 is the constant.

Operations performed on algebraic expression are:

**Addition & Subtraction:**

On performing addition or subtraction on algebraic expression the coefficients of same degree added or subtracted.

For example:=> 3x + 4x = 7x => a

^{3}+ 4a^{2}- 3a^{2}+ 2 = a^{3}+ a^{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) = 6a

^{2}- 24a + 6a - 24 = 6a^{2}- 18a - 24**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) => 9x

^{2}- 3x = 3x(3x - 1) => 4y^{2}+ 8y/ 2y+ 4 = 4y(y + 2)/ 2(y+2) = 4y/2 (where y ≠ 2 )**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}= x^{2}+ 2xy + y^{2}(x - y)^{2}= x^{2}- 2xy + y^{2}(x + y)^{3}= x^{3}+ 3x^{2}y + 3xy^{2}+ y^{3}(x - y)^{3}= x^{3}- 3x^{2}y + 3xy^{2}- y^{3}x^{2}- y^{2}= (x + y)(x - y) x^{3}- y^{3}= (x - y)(x^{2}+ xy + y^{2}) x^{3}+ y^{3}= (x + y)(x^{2}- xy + y^{2}) x^{2}+ y^{2}+ z^{2}= (x + y + z)^{2}- 2(xy + yz + zx) x^{3}+ y^{3}+ z^{3}- 3xyz = (x + y + z)(x^{2}+ y^{2}+ z^{2}- 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.

**Linear equation in one variable:**2a + 4 = 8

**Linear equation in two variables:**5a + 7b = 49

**A quadratic equation in one variable:**4a

^{2}+ 2a = 16