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 Algebra Basics for FPSC Tests | Algebraic Formulas Sheet


Algebra is the branch of mathematics that helps in the representation of problems or situations in the form of mathematical expressions. It involves variables like x, y, z, and mathematical operations like addition, subtraction, multiplication, and division to form a meaningful mathematical expression. All the branches of mathematics such as trigonometry, calculus, and coordinate geometry, involve the use of algebra. One simple example of an expression in algebra is 2x + 4 = 8.

Algebra deals with symbols and these symbols are related to each other with the help of operators. It is not just a mathematical concept, but a skill that all of us use in our daily life without even realizing it. Understanding algebra as a concept is more important than solving equations and finding the right answer, as it is useful in all the other topics of mathematics that you are going to learn in the future or you have already learned in past.

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What is Algebra?

Algebra is a branch of mathematics that deals with symbols and the arithmetic operations across these symbols. These symbols do not have any fixed values and are called variables. In our real-life problems, we often see certain values that keep on changing. But there is a constant need to represent these changing values. Here in algebra, these values are often represented with symbols such as x, y, z, p, or q, and these symbols are called variables. Further, these symbols are manipulated through various arithmetic operations of addition, subtraction, multiplication, and division, with an objective to find the values.

The above algebraic expressions are made up of variables, operators, and constants. Here the numbers 4 and 28 are constants, x is the variable, and the arithmetic operation of addition is performed.

Branches of Algebra

The complexity of algebra is simplified by the use of numerous algebraic expressions. Based on the use and the complexity of the expressions, algebra can be classified into various branches that are listed below:

  • Pre-algebra
  • Elementary Algebra
  • Abstract Algebra
  • Universal Algebra

Pre-algebra

The basic ways of presenting the unknown values as variables help to create mathematical expressions. It helps in transforming real-life problems into an algebraic expression in mathematics. Forming a mathematical expression of the given problem statement is part of pre-algebra.

Elementary Algebra

Elementary algebra deals with solving the algebraic expressions for a viable answer. In elementary algebra, simple variables like x, y, are represented in the form of an equation. Based on the degree of the variable, the equations are called linear equations, quadratic equations, polynomials. Linear equations are of the form, ax + b = c, ax + by + c = 0, ax + by + cz + d = 0. Elementary algebra based on the degree of the variables, branches out into quadratic equations and polynomials. A general form of representation of a quadratic equation is ax2 + bx + c = 0, and for a polynomial equation, it is axn + bxn-1+ cxn-2+ …..k = 0.

Abstract Algebra

Abstract algebra deals with the use of abstract concepts like groups, rings, vectors rather than simple mathematical number systems. Rings are a simple level of abstraction found by writing the addition and multiplication properties together. Group theory and ring theory are two important concepts of abstract algebra. Abstract algebra finds numerous applications in computer sciences, physics, astronomy, and uses vector spaces to represent quantities.

Universal Algebra

All the other mathematical forms involving trigonometry, calculus, coordinate geometry involving algebraic expressions can be accounted as universal algebra. Across these topics, universal algebra studies mathematical expressions and does not involve the study of models of algebra. All the other branches of algebra can be considered as the subset of universal algebra. Any of the real-life problems can be classified into one of the branches of mathematics and can be solved using abstract 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 complex variables like x2, x3, xn, xy, x2y, etc. Algebraic expressions are also known as polynomials. A polynomial is an expression consisting of variables (also called indeterminates), coefficients, and non-negative integer exponents of variables. Example: 5x3 + 4x2 + 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 ax2 + 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 ax3 + 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 is 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 to simplify large algebraic expressions. The exponential form represented as ax = n can be transformed into a logarithmic form as loga 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

 


Algebraic Formula Sheet


Below you can find all Algebraic Formulas Sheets for upcoming FPSC Tests. Click HERE to download the Formula sheet of Algebraic Expression for Upcoming tests of FPSC.

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.

Basic Rules and Properties of Algebra

The basic rules or properties of algebra for variables, algebraic expressions, or real numbers a, b and c are as given below,

  • Commutative Property of Addition: a + b = b + a

  • Commutative Property of Multiplication: a × b = b × a

  • Associative Property of Addition: a + (b + c) = (a + b) + c

  • Associative Property of Multiplication: a × (b × c) = (a × b) × c

  • Distributive Property: a × (b + c) = (a × b) + (a × c), or, a × (b – c) = (a × b) – (a × c)

  • Reciprocal: Reciprocal of a = 1/a

  • Additive Identity Property: a + 0 = 0 + a = a

  • Multiplicative Identity Property: a × 1 = 1 × a = a

  • Additive Inverse: a + (-a) = 0

 

Examples on Algebra

Example 1: Expand (2x + 3y)2 using the algebraic identities.Here we shall use an identity in algebra, (a + b)2 = a2 + 2ab + b2= 4x2 + 12xy + 9y2

Solution:

(2x + 3y)2 = (2x)2 + 2(2x)(3y) + (3y)2

Therefore, the answer is (2x + 3y)2 = 4x2 + 12xy + 9y2

Example 2: The present age of a person is double the age of his son. Ten years ago, his age was four times the age of his son. Use the concept of algebra and find the present age of the son.Let us consider the present age of the son as ‘x’ years. It is given that the age of the person is double the age of his son, so the age of the person is ‘2x’ years. Now considering the situation 10 years ago, the age of the son was (x – 10) years and the age of the person was (2x – 10) years. The question says that 10 years ago the age of the person was 4 times the age of his son. Therefore, this can be expressed as,2x – 10 = 4x – 40-2x = -30x = 30/2Therefore, the present age of the son is 15 years.

Solution:

x = 15

2x = 30

2x – 10 = 4(x – 10)

2x – 4x = -40 + 10


More on Algebra


What is Algebra?

Algebra is the branch of mathematics that represents problems in the form of mathematical expressions. It involves variables like x, y, z, and mathematical operations like addition, subtraction, multiplication, and division to form a meaningful mathematical expression.

How Many Types of Algebra are there?

The various types of algebra are elementary algebra, abstract algebra, linear algebra, boolean algebra, and universal algebra. These are named based on the problem we are able to solve through the use of algebra.

What is Abstract Algebra?

In algebra, which is a broad division of mathematics, abstract algebra, or modern algebra is the study of algebraic structures including groups, rings, fields, modules, vector spaces, lattices, and algebras.

What is the Highest Level of Algebra?

The highest level of algebra involves complex math topics of calculus, trigonometry, three-dimensional geometry, to name a few. Here algebra is used to represent complex problems and obtain the solutions for those problems.

What are the Basics of Algebra?

The basics of algebra include numbers, variables, constants, expressions, equations, linear equations, quadratic equations. Further, it involves the basic arithmetic operations of addition, subtraction, multiplication, and division within the algebraic expressions.

What are the Four Basic Rules of Algebra?

The four basic rules of algebra are the commutative rule of addition, commutative rule of multiplication, associative rule of addition, associative rule of multiplication. Commutative rule of addition: a + b = b + a, commutative rule of multiplication: a × b = b × a, associative rule of addition: a + (b + c) = (a + b) + c, associative rule of multiplication: a × (b × c) = (a × b) × c.

What is the Fundamental Theorem of Algebra?

The fundamental theorem of algebra states that an algebraic expression of n degree has n roots. An algebraic expression of the form f(x) = xn has n roots as answers.

What is the Easiest Way to Learn Algebra?

The easiest way to learn algebra is to know the three basics of problem representation and solving. First, the problem statement should be represented in the form of a solvable equation. Secondly, the manipulation of the values by moving the numbers across the equals to sign should be performed with ease. Third, the arithmetic operations like addition, subtraction, multiplication, and division should be performed proficiently.

How is Algebra Used in Daily Life?

Algebra helps to find the values of unknown quantities in our daily life. The unknown quantities are represented as variables x, y in the form of an equation. Further, the equations involving arithmetic operations are solved to find the values of those variables. Quantities like speed, time, distance, currencies can be represented as variables in algebra.

How do you Solve Basic Algebra?

Solving the algebraic expressions involves three simple steps. First, identify and group the variables of the same kind. Second, transform the variable on one side and the constants on another side. Then, bring all the variables of a similar kind to one side. And at last, perform the needed arithmetic operations.

What are the Basic Operations in Algebra?

The four basic operations in algebra are addition, subtraction, multiplication, and division. Different operators (+, -, ×, /) are used to separate different terms to perform these operations among the operands.