In this article, we will discuss a part of mathematics that involves variables. We will discuss algebra, algebraic expressions, equations and inequalities.

Contents

## Introduction to Algebra

Algebra in very simple terms is part of mathematics that involves variables.

A **variable** is a letter that represents either a specific number or all numbers.

In algebraic expressions, the variable represents either all numbers or all numbers with a few specific exceptions.

In algebraic equations, the variable represents one or more initially unknown values, and the goal is to solve for those specific values.

Now that we know what a variable is, let’s define some additional terms as used in algebra.

### Terminologies in Algebra

##### Constant

A constant is a number or a symbol such as π that doesn’t change in value.

##### Term

A term is a product of constants and variables including powers of variables e.g 2, x, 2x^{2}, z^{4}, x^{2}y^{3}z^{4 }etc.

##### Coefficient

A coefficient is the constant factor of a term e.g in 3x^{2}, 3 is the coefficient.

When no constant is written e.g in x^{2}, the coefficient is 1.

##### Expression

An expression is a collection of one or more terms joined by addition or subtraction e.g.

x+y, x^{2}-y^{2}, 1+x^{3}+x^{6}, x^{2}+2xy+y^{2 }etc.

Expressions don’t have equal signs.

##### Monomial

A monomial is an expression with exactly one term e.g.

2x^{2}, 12, z, x^{2}y^{3 }etc.

##### Binomial

A binomial is an expression with exactly two terms e.g.

x+5, x^{2}+y^{2}, a+b etc.

##### Trinomial

A trinomial is an expression with exactly three terms e.g x^{2}+2xy+y^{2}.

##### Polynomial

A polynomial is an expression with any number of terms involving only one variable.

##### Equations

An equation is simply an expression with equal sign e.g x^{2}+2xy+y^{2 }= 0.

##### Linear

A term with a single power of a variable (that is no explicit exponent).

##### Quadratic

A quadratic is a term with the square of a single variable.

Sometimes the words linear and quadratic can describe individual terms, but they can also describe entire expressions involving a single variable.

In a linear expression, the highest power of the variable is 1.

In a quadratic expression, the highest power of the variable is 2.

##### Cubic

A term with the cube of a single variable.

## Algebraic Expressions

We can simplify algebraic expressions by adding or subtracting like terms e.g (x^{3}-3x^{2}) + 2x^{2}+5x^{3} = x^{3}+5x^{3}-3x^{2}+2x^{2} = 6x^{3}-x^{2}.

When adding expression with parentheses, we can simply remove the parentheses and perform the addition directly e.g 2x+(4x+5y) = 2x+4x+5y = 6x+5y but when subtracting an expression in parentheses, we have to change each term to its opposite sign when we remove the parentheses e.g 2x-(4x+5y) = 2x-4x-5y = -2x-5y.

### The FOIL Method

Let us now discuss multiplication of two binomial expressions each of which involves addition or subtraction in the form (a+b)(c+d).

We can use the distributive law in this case but there is a very convenient shortcut summarized by the mnemonic FOIL.

F = First (**a**+b)(**c**+d) [a*c]

O = Outer (**a**+b)(c+**d**) [a*d]

I = Inner (a+**b**)(**c**+d) [b*c]

L = Last (a+**b**)(c+**d**) [b*d]

The product of the binomials is the sum of those four individual products.

Let us look at an example (2x+y)(x+2y):

F = First 2x*x

O = Outer 2x*2y

I = Inner y*x

L = Last y*2y

(2x+y)(x+2y) = 2x^{2} + 4xy + xy + 2y^{2 }= 2x^{2} + 5xy + 2y^{2 }