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Algebra, Equations and Inequalities

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

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, 2x2, z4, x2y3zetc.

Coefficient

A coefficient is the constant factor of a term e.g in 3x2, 3 is the coefficient.

When no constant is written e.g in x2, the coefficient is 1.

Expression

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

x+y, x2-y2, 1+x3+x6, x2+2xy+yetc.

Expressions don’t have equal signs.

Monomial

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

2x2, 12, z, x2yetc.

Binomial

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

x+5, x2+y2, a+b etc.

Trinomial

A trinomial is an expression with exactly three terms e.g x2+2xy+y2.

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  x2+2xy+y= 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 (x3-3x2) + 2x2+5x3 = x3+5x3-3x2+2x2 = 6x3-x2.

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) = 2x2 + 4xy + xy + 2y= 2x2 + 5xy + 2y

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