ALGEBRA
ALGEBRA TOPICS
Algebra is divided into numerous topics to help for a detailed study. Here, we have listed some
of the important topics of algebra such as algebraic expressions and equations, sequence and
series, exponents, logarithm, and sets.
ALGEBRAIC EXPRESSIONS
An algebraic expression in algebra is formed using integer constants, variables, and basic
arithmetic operations of addition(+), subtraction(-), multiplication(×), and division(/). An
example of an algebraic expression is 5x + 6. Here 5 and 6 are fixed numbers and x is a
variable. Further, the variables can be simple variables using alphabets like x, y, z or can have
2 3 n 2
complex variables like x , x , x , xy, x y, etc. Algebraic expressions are also known as
polynomials. A polynomial is an expression consisting of variables (also called indeterminates),
3 2
coefficients, and non-negative integer exponents of variables. Example: 5x + 4x + 7x + 2 = 0.
An equation is a mathematical statement with an 'equal to' symbol between two algebraic
expressions that have equal values. Given below are the different types of equations, based on
the degree of the variable, where we apply the concept of algebra:
● Linear Equations: Linear equations help in representing the relationship between
variables such as x, y, z, and are expressed in exponents of one degree. In these
linear equations, we use algebra, starting from the basics such as the addition and
subtraction of algebraic expressions.
● Quadratic Equations: A quadratic equation can be written in the standard form as ax 2
+ bx + c = 0, where a, b, c are constants and x is the variable. The values of x that
satisfy the equation are called solutions of the equation, and a quadratic equation
has at most two solutions.
● Cubic Equations: The algebraic equations having variables with power 3 are referred
to as cubic equations. A generalized form of a cubic equation is ax 3 + bx2 + cx + d =
0. A cubic equation has numerous applications in calculus and three-dimensional
geometry (3D Geometry).
SEQUENCE AND SERIES
A set of numbers having a relationship across the numbers is called a sequence. A sequence is
a set of numbers having a common mathematical relationship between the number, and a
series is the sum of the terms of a sequence. In mathematics, we have two broad number
sequences and series in the form of arithmetic progression and geometric progression. Some of
these series are finite and some series are infinite. The two series are also called arithmetic
progression and geometric progression and can be represented as follows.
● Arithmetic Progression: An Arithmetic progression (AP) is a special type of
progression in which the difference between two consecutive terms is always a
constant. The terms of an arithmetic progression series are a, a+d, a + 2d, a + 3d, a
+ 4d, a + 5d, .....
● Geometric Progression: Any progression in which the ratio of adjacent terms is
fixed is a Geometric Progression. The general form of representation of a geometric
sequence is a, ar, ar2, ar3, ar4, ar5, .....
EXPONENTS
Exponent is a mathematical operation, written as an. Here the expression an involves two
numbers, the base 'a' and the exponent or power 'n'. Exponents are used to simplify algebraic
expressions. In this section, we are going to learn in detail about exponents including squares,
cubes, square root, and cube root. The names are based on the powers of these exponents.
The exponents can be represented in the form an = a × a × a × ... n times.
LOGARITHMS
The logarithm is the inverse function to exponents in algebra. Logarithms are a convenient way
x
to simplify large algebraic expressions. The exponential form represented as a = n can be
transformed into logarithmic form as log
a
a
n = x. John Napier discovered the concept of Logarithms in 1614. Logarithms have now become
an integral part of modern mathematics.
SETS
A set is a well-defined collection of distinct objects and is used to represent algebraic variables.
The purpose of using sets is to represent the collection of relevant objects in a group. Example:
Set A = {2, 4, 6, 8}..........(A set of even numbers), Set B = {a, e, i, o, u}......(A set of vowels).
ALGEBRAIC FORMULAS
An algebraic identity is an equation that is always true regardless of the values assigned to the
variables. Identity means that the left-hand side of the equation is identical to the right-hand
side, for all values of the variables. These formulae involve squares and cubes of algebraic
expressions and help in solving the algebraic expressions in a few quick steps. The frequently
used algebraic formulas are listed below.
● (a + b)2 = a2 + 2ab + b2
● (a - b)2 = a2 - 2ab + b2
● (a + b)(a - b) = a2 - b2
● (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
● (a + b)3 = a3 + 3a2b + 3ab2 + b3
● (a - b)3 = a3 - 3a2b + 3ab2 - b3
Let us see the application of these formulas in algebra using the following example,
Example: Using the (a + b)2 formula in algebra, find the value of (101)2.
Solution:
Given: (101)2 = (100 + 1)2
Using algebra formula (a + b)2 = a2 + 2ab + b2, we have,
(100 + 1)2 = (100)2 + 2(1)(100) + (1)2
(101)2 = 10201
For more formulas check the page of algebraic formulas, containing the formulas for expansion
of algebraic expressions, exponents, and logarithmic formulas.
ALGEBRAIC OPERATIONS
The basic operations covered in algebra are addition, subtraction, multiplication, and division.
● Addition: For the addition operation in algebra, two or more expressions are
separated by a plus (+) sign between them.
● Subtraction: For the subtraction operation in algebra, two or more expressions are
separated by a minus (-) sign between them.
● Multiplication: For the multiplication operation in algebra, two or more expressions
are separated by a multiplication (×) sign between them.
● Division: For the division operation in algebra, two or more expressions are
separated by a "/" sign between them.
