Algebra for Aptitude

1.   NUMBER SYSTEM

♦ EUCLID’S DIVISION LEMMA:

Euclid was the first Greek Mathematician who gave a new way of thinking the study of geometry. He also made important contributions to the number theory. Euclid’s Lemma is one of them. It is a proven statement which is used to prove other statements.

Let ‘a’ and ‘b’ be any two positive integers. Then, there exist unique integers q and r such that

a = bq + r  ; 0 <= r

Now, we say ‘a’ as dividend, ‘b’ as divisor, ‘q’ as quotient and ‘r’ as remainder.

Dividend = (divisor x quotient) + remainder

Theorem (1):    Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.

Theorem (2):    Let x = p/q be a rational number, such that the prime factorization of q is of the form 2ⁿ5^m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

Theorem (3):    Let x = p/q be a rational number, such that the prime factorization of q is not of the form 2ⁿ5^m, where n, m are non-negative integers. Then, x has a decimal expansion which is not-terminating repeating (recurring).

Hence, we can say that the decimal expansion of every rational number is either terminating or non-terminating repeating (recurring).

• Natural Numbers: All natural numbers are said to be the counting numbers. In set notation, Natural numbers are denoted by N. N = {1, 2, 3, 4, 5, …..}
• Whole Numbers: All natural numbers together with 0 (zero) form the set of all whole numbers.

Thus, W = {0, 1, 2, 3, 4, 5, …..} is the set of all whole numbers.

The set of whole numbers are represented by W.

The whole numbers and natural numbers can be represented on the number line, as shown:

• Integers: All whole numbers with their absolutes form the set of all whole numbers.

Thus, W = {…., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …..} is the set of all integers. The set of Integers are represented by Z or I.

♦ PROPERTIES OF INTEGERS:

(i)    The number line goes on forever in both directions. This is indicated by the arrows.

(ii)   Whole numbers greater than zero are called positive integers. These numbers are to the right of zero on the number line.

(iii)  Whole numbers less than zero are called negative integers. These numbers are to the left of zero on the number line.

(iv)  The integer zero is neutral. It is neither positive nor negative.

(v)   The sign of an integer is either positive (+) or negative (-). except zero, which has no sign.

(vi)  Two integers are opposites if they are each the same distance away from zero, but on opposite sides of the number line. One will have a positive sign, the other a negative sign. In the number line above.+3 and – 3 are labeled as opposites.

• Rational Numbers:

The numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0, are called rational numbers. The set of all rational numbers are represented by Q.

• Equivalent Rational Numbers:

Let p/q be a rational numbers and r be any integer, then we have: p/q = rp/rq        

The rational numbers p/q and rp/rq are equivalent rational numbers.

Thus, 3/5, 6/10, 9/15, 12/20 etc. are all equivalent rational numbers.

• Operations on Rational Numbers:

For any two rational numbers p/q and m/n, we define:

(i) Addition: p/q + m/n = pn+mq/qn

(ii) Subtraction: p/q – m/n = pn-mq/qn

(iii) Multiplication: p/q x m/n = pm/nq

(iv) Division: p/q ÷ m/n = p/q x n/m , where q ≠ 0, n ≠ 0 .

• Properties of Rational Numbers:

1.    (i)    The sum of two rational numbers is always rational
[Closure property for Addition]

(ii)   The product of two rational numbers is always rational.
[Closure property for Multiplication]

2.    For any two rational numbers p/q and m/n, we have:
(i)    p/q + m/n = m/n + p/q                [Commutative Law of Addition]
(ii)   p/q x m/n = m/n x p/q                [Commutative Law of Multiplication]

3.    For any three rational numbers p/q, m/n, x/y we have:
(i)    ( p/q + m/n) + x/y = p/q + (m/n + x/y)         [Associative Law of Addition]
(ii)    ( p/q x m/n) x x/y = p/q x (m/n x x/y)         [Associative Law of Multiplication]

4.    The difference of two rational numbers is always rational.

5.    (i)    We have 0 ε Q such that p/q + 0 = p/q = 0 + p/q for any rational number p/q.
[0 is the Additive Identity]

(ii)   We have 1 ε Q such that p/q x 1 = p/q = 1 x p/q for any rational number p/q.
[1 is the Multiplicative Identity]

6.    (i)    For each rational number p/q we have a rational number (-p/q) such that
p/q + (-p/q) = (-p/q) + p/q = 0. [(-p/q) is called the Additive Inverse of p/q]

(ii)   For every non-zero rational number p/q we have a rational number p/q such that p/q x q/p = 1. [p/q is called the Multiplicative Inverse of p/q]               

¤ Note :Zero does not have its multiplicative inverse.

• Insertion of Rational Numbers between Two Rational Numbers:

If a and b are two rational numbers, then 1/2(a+b) is also a rational number and lies between a and b. Actually, between any two rational numbers, there exist an infinite number of rational numbers.

♦ IRRATIONAL NUMBERS

Real numbers which are not rational number are called irrational numbers. For example, real numbers like √2 which are not rational are categorized as irrational. When irrational numbers are expressed as decimals, those numbers are non-terminating and non-recurring. That is, Irrational numbers cannot be denoted in the form of p/q where p,q ε  Z and q ≠ 0. 

A number which when expressed as a decimal that is non-terminating and non-recurring (non repeating) is called an irrational number. Irrational numbers are denoted by Q. 
A square root of every non-perfect square is an irrational number and similarly, cube roots of non-perfect cubes are also examples of irrational numbers.

• Some famous irrational numbers

(i)    Pi (π) is a famous irrational number. The value of Pi can be calculated to over one million decimal places and still there is no pattern. The first few digits are looking as: 3.1415926535897932384626433832795 ……

(ii)   The number e (Euler’s Number) is also a famous irrational number. The value for e has been calculated to lots of decimal places without any pattern showing. Some of first few digits look like this: 2.7182818284590452353602874713527 ……

(iii)  The Golden Ratio is another irrational number. The first few digits of Golden ratio is 1.61803398874989484820…

Not all the square roots, cube roots are irrational but many of them are irrational numbers.

• Some Important Results on Irrational Numbers:

1.    The negative of an irrational number is an irrational number. e.g., the negative of the irrational number  is -√2 which is also irrational.

2.    The sum of a rational and an irrational number is an irrational number e.g. 2 + √3 is an irrational number.

3.    The sum of two irrational numbers may be either irrational or rational.

e.g. (i) 3√2 + 4√2 = (3 + 4)√2 = 7√2, which is irrational.

(ii) (2 + √3) + (4 – √3) = 2 + 4 = 6, which is rational.

4.    The product of a non-zero rational number with an irrational number is always an  irrational number.
e.g. 3√2 is an irrational number.

♦ REAL NUMBERS:

The totality of all rational and irrationals forms the set R of all real numbers

• Properties of Real Numbers:

1.    (i) The sum of two real numbers is always a real number.
[Closure property for addition]

(ii) The product of two real numbers is always a real number.
[Closure property for multiplication]

2.    For any two real numbers a and b, we have:
(i) a + b = b + a                               [Commutative law of addition]
(ii) a x b = b x a                                 [Commutative law of multiplication]

3.    For any three real numbers a, b, c we have:
(i) (a + b) + c = a + (b + c)                [Associative law of addition]
(ii) (a x b) x c = a x (b x c)                 [Associative law of multiplication]

4.    For any three real numbers a, b, c, we have:
a(b + c) = ab + bc and (a + b) c = ac + bc
[Distributive laws of multiplication over addition]

5.    If  a ≠ 0, then 1/a is called the reciprocal of a and we have : a x 1/a = 1.

Prime Factors: A factor of a given number is called a prime factor if this factor is a prime number.Example: The factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. Out of these 2, 3 and 7 are prime numbers. Therefore, 2, 3 and 7 are the prime factors of 42.

Common Factors: A number which divides each one of the given numbers exactly, is called a common factor of each of the given numbers.Example: 4 divides each one of 212 and 356 exactly. Therefore, 4 is a common factor of 212 and 356.

• H.C.F. (HIGHEST COMMON FACTOR) OR G.C.D. (GREATEST COMMON DIVISOR)

H.C.F. or G.C.D. of two or more numbers is the greatest number that divides each one of them exactly.

• Methods of Finding the H.C.F. of Given Numbers

       Prime Factorization Method :

Suppose we have to find the H.C.F. of two or more numbers.

Step 1.    Express each one of the given numbers as the product of prime factors.
Step 2.    The product of terms containing least powers of common prime factors gives the H.C.F. of the given numbers.

• L.C.M (Least Common Multiple)

The L.C.M. of two or more natural numbers is the least natural number which is a multiple of each of the given numbers.

• Methods of Finding the L.C.M. of Given Numbers

       Prime Factorization Method :
Suppose we have to find the L.C.M. of two or more numbers.

Step 1.    Express each one of the given numbers as the product of prime factors.
Step 2.    The product of all the different prime factors each raised to highest power that appears in the prime factorization of any of the given numbers, gives the L.C.M. of the given numbers.

Relation between H.C.F. and L.C.M. of two numbers: Product of two given numbers = Product of their H.C.F. and L.C.M.

Squares:  The square of a number is equal to number raised to the power 2.

Factors and Multiples: If a number ‘a’ divides another number ‘b’ exactly, we say that ‘a’ is a factor of ‘b’ and ‘b’ is a multiple of ‘a’.Thus, a factor of a number is an exact divisor of that number.And, a number is said to be a multiple of any of its factors.

 

Even Numbers:  All the multiples of 2 are called even numbers.

 

Odd Numbers: Numbers which are not multiples of 2 are called odd numbers.

 

Prime Numbers: Each of the numbers which has exactly two factors, namely, 1 and the number itself, is called a prime number.

Composite Numbers: Numbers having more than two factors are known as composite numbers.

♦ TESTS OF DIVISIBILITY

•        Test of Divisibility by 2:

A number is divisible by 2, if its units digit is any of the digits 0, 2, 4, 6 and 8.

•        Test of Divisibility by 3:

A number is divisible by 3, if the sum of its digits is divisible by 3.

•       Test of Divisibility by 4:

A number is divisible by 4, if the number formed by its last two digits is divisible by 4.

•    Test of Divisibility by 5:

A number is divisible by 5, if its units digit is either 0 or 5.

•   Test of Divisibility by 6:

A number is divisible by 6, if it is divisible by 2 as well as 3.

• Test of Divisibility by 8:

A number is divisible by 8, if the number formed by its last three digits is divisible by 8.

• Test of Divisibility by 9:

A number is divisible by 9, if the sum of its digits is divisible by 9.

• Test of Divisibility by 10:

A number is divisible by 10, if its unit’s digit is 0.

• Test of Divisibility by 11: A number is divisible by 11, if the difference between the sum of its digits at odd places and sum of the digits at even places is either 0 or a number divisible by 11.

♦ PERFECT SQUARE OR SQUARE NUMBER:​

A natural number n is called a perfect square or a square number if there exists a natural number m such that n = m².

In other words, a natural number is a perfect square if it is the square of some natural number.

Since 4 = 2 × 2 = 2². Therefore, 4 is a square number.

Also,      25 = 5 × 5 = 5², 25 is a square number or a perfect square.

144 = 12 × 12 = 12², so 144 is a square number or a perfect square etc.

♦ PROPERTIES OF SQUARE NUMBERS:​

Property 1:   A number having 2, 3, 7 or 8 at unit’s place is never a perfect square. In other words, no square number ends in 2, 3, 7 or 8.

Property 2:   The number of zeros at the end of a perfect square is always even. In other words, a number ending in an odd number of zeroes is never a perfect square.

Property 3:   Square of even numbers are always even numbers and squares of odd numbers are always odd numbers.

Property 4:   The Square of a natural number other than one is either a multiple of 3 or exceeds a multiple of 3 by 1.
                  In other words, a perfect square leaves remainder 0 or 1 on division by 3.

Property 5:   The Square of a natural number other than one is either a multiple of 4 or exceeds a multiple of 4 by 1.
                  In other words, a perfect square leaves remainder 0 or 1 on division by 4.

Property 6:   The unit’s digit of the square of a natural number is the unit’s digit of the square of the digit at unit’s place of the given natural number.

♦   PYTHAGOREAN TRIPLETS:

A triplet (m, n, p) of three natural numbers m, n and p is called a Pythagorean triplet, if m² + n² = p²
Property 7:   For any natural number m greater than 1, (2m, m² – 1, m² + 1) is a Pythagorean triplet.

♦   PERFECT CUBES

A natural number is said to be a perfect cube, if it is the cube of some natural number.
In other words, a natural number n is a perfect cube if there exists a natural number m whose cube is n i.e., n = m³

♦ SOME PROPERTIES OF CUBES OF NATURAL NUMBERS

The cubes of natural numbers have the following interesting properties:

Property 1:     Cubes of all even natural numbers are even

Property 2:     Cubes of all odd natural numbers are odd

Property 3:     The sum of the cubes of first n natural numbers is equal to the square of their sum. That is,
                     1³ + 2³ + 3³+ …. + n³ = (1 + 2 + 3 + …. n)²

Property 4:     Cubes of the numbers ending in digits 1, 4, 5, 6 and 9 are the numbers ending in the same digit. Cubes of numbers ending in digit 2 ends in digit 8 and the cube digits 3 and 7 ends in digit 7 and 3 respectively.

♦ CUBE ROOT OF A RATIONAL NUMBER

If x and a are two rational numbers such that x³ = a, then we say that x is the cube root of a and we write ³√a = x.

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2. POLYNOMIALS

♦  POLYNOMIALS IN ONE VARIABLE:

The algebraic expressions having only one variable like 2x, 3x, –x, –1/2x etc.

In the above algebraic expressions (a constant) × (a variable).

i.e.  2(a constant) × x(a variable), 3(a constant) × x(a variable), –1(a constant) × x(a variable), –1/2(a constant) × x(a variable)

(i)    Variable:
A symbol has any real value is called variable.
E.g. x, y, z … are variables.

(ii)   Constant:
A symbol with a fixed numerical value is called constant.
e.g. –1, –5, 2, 3, 4, –5/2, 3/4 …………………… are constants.

(iii)  Terms:
In a polynomial 5x² + 3x + 7, the expression 5x², 3x and 7 are called the terms of the polynomial.
Here, the number of terms is 3. But, the number of terms in 3x³ + 2x² – 6x + 10 is 4

(iv)  Coefficient:
Each term of a polynomial has a coefficient. Hence, 3x³ + 2x² – 6x + 10, the coefficient of x³ is 3, the coefficient of x² is 2, the coefficient of x is –6 and 10 is the coefficient of x⁰ (∵ x⁰ = 1)

(v)   Zero polynomial:
A polynomial consisting of one term, namely zero only, is called a zero polynomial. The degree of zero polynomial is not defined.

♦  SPECIAL NAME OF POLYNOMIALS:

(i)    Monomial: A polynomial contains only one term is called monomial, (mono = one)
Example: 3, 5x, 6xy, 3x²y, ……………. etc.

(ii)   Binomial: A polynomial contains two terms is called binomial. (Binomial = two)
Example: 5x + 2, 3x + y, xyz – 2y, 5x² + 2y³ ………. etc.

(iii)  Trinomial: A polynomial contains three terms is called trinomial (Tri = three)
Example: 5x + 2y – 3z, x³ – 2x² + 3x, y⁴ + 2x³ + z ……………..etc.

♦  DEGREE OF A POLYNOMIAL:

If p(x) is a polynomial in x, then the highest power of x in p(x) is called the degree of the polynomial p(x).
The degree of a non-zero constant polynomial is zero.

♦  DIFFERENT TYPES OF POLYNOMIALS:

(i)    Linear polynomials:
A polynomial of degree one is called a linear polynomial.Thegeneral form of a linear polynomial in x is of the form f(x) = ax +b, where a and b are real numbers and a ≠ 0.

(ii)   Quadratic polynomials:
A polynomial of degree two is called a quadratic polynomial.The name ‘quadratic’ has been derived from the word ‘quadrate’ means ‘square’. Thegeneral form of a quadratic polynomial in x is of the form f(x) = ax² + bx +c, where a, b, c are real numbers and a ≠ 0.

(iii)  Cubic polynomials:
A polynomial of degree three is called a cubic polynomial.The general form of a cubic polynomial is ax³ + bx² + cx + d, where a, b, c, d are real numbers and a ≠ 0.

♦  THE VALUE OF A POLYNOMIAL:

If p(x) is a polynomial in x, and if K is any real number, then the value obtained by replacing x by K in p(x), is called the value of p(x) at x = k, and is denoted by p(k).
Let us find the value of  p(x) = x² – 3x – 4 ; if x = –1.

P(–1) = (–1)² – 3 (–1) – 4

= 1 + 3 – 4 = 4 – 4

♦  ZEROS OF A POLYNOMIAL:

A real number K is said to be a Zero of a polynomial p(x), if p(K) = 0,
E.g. Let    p(x)    = x² – 4x + 4
If p(x) at x = 2, then
P(2)  = (2)² – 4(2) + 4
= 4 – 8 + 4 = 8 – 8 = 0
Hence, we can say that K = 2 is a zero of polynomial p(x).

♦  PROPERTIES OF THE ZEROS OF A POLYNOMIAL:

(i)   Every real number is a zero of the zero polynomial
(ii)  A zero of a polynomial need not be 0
(iii) 0 may be a zero of a polynomial
(iv) Every linear polynomial has one and only one zero
(v)  A polynomial can have more than one zero.

♦  RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF POLYNOMIAL:

Generally, Let α and ß be the zeroes of the quadratic polynomial p(x) = ax² + bx + c, a ≠ 0, then we know that (x – α) and (x – ß) are the factors of p(x).
Hence,     ax² + bx + c = K  (x – α) (x – ß) ; Where K = a constant.

= K [x² -(α + ß)x + αß]

= Kx² – K(α + ß)x + K αß

Now, comparing the coefficients of x², x and constant terms on both sides, we get
a = K, b = – K (α + ß) and c = Kαß  

   α + ß = -b/a, αß = c/a   

i.e.,   ∴   Sum of the zeroes is α + ß = -b/a = -(coefficient of x)/coefficient of x²

Product of zeroes is αß = c/a = Constant term/coefficient of x²

Similarly, if α, ß and Y are the zeroes of the cubic polynomial, ax³ + bx² + cx + d, then
(i)  α + ß + Y = -b/a
(ii) αß + ßY + Yα = c/a
(iii) αßY = -d/a
(iv) p (x) = { x³ – (α + ß + Y)x² + (αß + ßY + Yα)x – αßY}

♦  DIVISION ALGORITHM FOR POLYNOMIALS:

Dividend = Divisor × Quotient + Remainder

♦ REMAINDER THEOREM:

Let p(x) be any polynomial of degree greater than or equal to one and ‘a’ be any real number. If p(x) is divided by the linear polynomial x – a, then the remainder is p(a).

♦ DIVISION OF A POLYNOMIAL:

Let polynomial p(x) be divided by another polynomial g(x), we get quotient q(x) and remainder r(x).

• Dividend = (Divisor × Quotient) + Remainder

I.e. p(x) = g(x) . q(x) + r(x)

♦  FACTORIZATION OF POLYNOMIALS:

       Factor Theorem:
If p(x) is a polynomial of degree n > 1 and a is any real number, then

(i)   x – a is a factor of p(x); if p(a) = 0, and

(ii)  p(a) = 0; if x – a is a factor of p(x).

♦ FACTORIZATION OF THE POLYNOMIAL ax² + bx + c BY SPLITTING THE MIDDLE TERM:

Let x + p and x + q be two linear factors of the polynomial ax² + bx + c

   x² + bx + c = (x + p) (x + q)

=>  x² + bx + c = x² + (p + q)x + pq
Now, comparing the coefficients of like power (exponent) of x on both sides, we get

b = p + q, c = pq

To factorize the given quadratic polynomial, we have to find two numbers p and q such that p + q = b and pq = c.

♦ ALGEBRAIC IDENTITIES:

(i)  (a + b)² = a² + 2ab + b²
(ii) (a – b)² = a² – 2ab + b²
(iii) a² – b² = (a + b) (a – b)
(iv) (x + a) (x + b) = x² + (a + b)x + ab
(v)  (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
(vi) (a – b – c)² = a² + b² + c² – 2ab + 2bc – 2ca
(vii) (a + b)³ = a³ + 3a²b + 3ab² + b³
(a + b)³ = a³ + 3ab(a + b) + b³
(viii) (a – b)³ = a³ – 3a_²b + 3ab² – b³
(a – b)³ = a³ – 3ab(a – b) – b³
(ix) a³ + b³ = (a + b) (a² – ab + b²)
(x)  a³ – b³ = (a – b) (a² + ab + b²)
(xi) a³ + b³ + c³ – 3ab = (a + b + c) (a² + b² + c² – ab – bc – ca)
(xii) If a + b + c = 0, then a³ + b³ + c³ = 3abc

• Laws of exponents for real numbers:

(i) a^m . aⁿ = a^m+n  e.g. x⁵ . x⁷ = x⁵⁺⁷ = x¹²  
(ii) (a^m)ⁿ = a^mn e.g. [x³]⁶ = x³⁺⁶ = x¹⁸
(iii) a^m/aⁿ = a^m-n (when m > n)    e.g. a¹⁵/a⁸ = a^15-8 = a⁷
(iv) a^m b^m = (ab)^m e.g. x⁹ y⁹ = (xy)⁹
(v)  a^-m = 1/a^m e.g. x^-10 = 1/x¹⁰
(vi) a⁰ = 1 e.g. x⁰ = 1
(vii) √a = a^1/2 e.g. √x = x^1/2
(viii) (a) ⁿ√2 = a^1/n
e.g. ³√x = a^1/3, where n is any positive number and a (rational number) > 0
(b) ⁿ√a^m = a^m/n e.g.  ³√x² = x^2/3 .

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3. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

♦  PAIR OF LINEAR EQUATIONS IN TWO VARIABLES:

The general form of a pair of linear equations in two variables x and y is
a₁x + b₁y + c₁ = 0                                     … (i)
And  a₂x + b₂y + c₂ = 0                                     … (ii)
Where a₁, b₁, c₁, a₂, b₂, c₂ are all real numbers and a²₁ + b²₁ ≠ 0, a²₂ + b²₂ ≠ 0.

A pair of linear equations in two variables can be represented by two straight lines. These are the following three possibilities of them.

• The two lines will intersect at a point
• The two lines will not intersect, i.e., they are parallel and
• The two lines will be coincident.

       We can show these three possibilities in the following figures:

A pair of linear equations can be represented by both ways (i.e., algebraic and geometric ways).

The lines representing a pair of linear equations in two variables and the existence of solutions are follows:

(i)    In this case the lines will intersect in a single point and the pair of equations have a unique solution.
(ii)   In this case, the lines will be parallel and the pair of equations will have no solution.
(iii)  In this case, the lines will be coincident and the pair of equations will have infinitely many solutions.

If the lines represented by the equation

a₁x + b₁y + c₁ = 0

And  a₂x +b₂y + c₂ = 0

(i)    Intersecting, then a₁/a₂ ≠ b₁/b₂
(ii)   Coincident, then a₁/a₂ = b₁/b₂ = c₁/c₂ and
(iii)  Parallel, then a₁/a₂ = b¹/b² ≠ c₁/c₂

♦  ALGEBRAIC METHODS OF SOLVING A PAIR OF LINEAR EQUATIONS :

There are several alternate algebraic methods by which we can solve the solutions.

       (a)  Substitution Method:

       (b)  Elimination Method:

       (c) Cross – Multiplication Method:

If       a₁x + b₁y + c₁ = 0                       … (i)
a₂x + b₂y + c₂ = 0                       … (ii)

Then,

We can remember the above formula by the following diagram

4. QUADRATIC EQUATIONS

♦  METHODS OF SOLVING QUADRATIC EQUATIONS:

       By Factorization: When an algebraic expression can be written as the product of two or more expressions, then each of these expressions is called a factor of the given expression.

• G.C.F. or H.C.F of Monomials:

The greatest common factor of given monomials is the common factor having greatest coefficient and highest power of the variables.

       G.C.F. or H.C.F of Monomials = (G.C.F. or H.C.F of numerical coefficients) x ( G.C.F. or H.C.F of literal coefficients)

♦  FACTORISATION BY TAKING OUT THE COMMON FACTOR:

       Case I: When each term of the given expression contains a common monomial factor.

       Method: We proceed stepwise, as under:

       Step1. Find the H.C.F. of all the terms of the given expression.

       Step2. Divide each term of the given expression by the H.C.F. so obtained. Enclose the quotients within a bracket and keep the common monomial outside the bracket.

       Case II. When a polynomial is a common multiplier of each term of the given expression. In this case we take out the common multiplier and use the distributive law.

♦ FACTORISATION BY GROUPING THE TERMS:

Sometimes in a given expression it is not possible to take out a common factor directly. In such cases, we have to make a suitable arrangement of the terms and group them in such a manner as to have a common polynomial. This can now be factorised as discussed above.

♦ FACTORISING THE DIFFERENCE OF TWO SQUARES:

In this case, we use the formula(a² – b²) = (a+b) (a-b).

♦ FACTORISATION OF TRINOMIAL OF THE FORM  (x² + px + q):

Let us find two numbers a and b such that a + b =p and ab = q. Then,
x² + px + q = x² + (a + b)x + ab
x² + px + q = x² + ax + bx + ab                      
x² + px + q = x(x + a) + b(x + a)      
x² + px + q = (x + a) (x + b)      
hence x² + px + q = (x + a) (x + b)      

♦ FACTORISATION OF TRINOMIALS OF THE FORM: (Ax² + Bx+ C)

Let us find two numbers a and b such that (a + b) = A and ab = AC.
We can now proceed in the above case.
The following examples will make the ideas more clear.
∴ 8 – 18x -5x² = (2 – 5x)(4 + x)

       2. Bycompleting the square:

       3. ByQuadratic Formula Method:

ax² + bx + c = 0; a ≠ 0
=> x² + (b/a)x + c/a = 0                                [Dividing both sides by a]
=>  (x²) + 2 (b/2a).(x) + (b/2a)²-(b/2a)² + c/a = 0
=>  (x + b/2a)² – (b/2a)² + c/a = 0
=>  (x + b/2a) – b²/4a² + c/a = 0
=>  (x + b/2a) -(b_²-4ac)/4a² = 0
=>  (x + b/2a) = (b_²-4ac)/4a²                          … (i)
If b2 – 4ac > 0; then by taking the square roots both sides in (i), we get
x +  b/2a = + √b² – 4ac/ 2a

=> x = -b/2a + √b² – 4ac/ 2a

∴ x =  (-b + √b2 – 4ac)/ 2a

Hence, the roots of ax² + bx + c = 0 are
 α = (-b + √b² – 4ac)/ 2a
and   ß = (-b – √b² – 4ac)/ 2a

♦  NATURE OF ROOTS:

In quadratic equation ax² + bx + c = 0; a ≠ 0
Then
(i)  If b² – 4ac > 0; then it has two distinct real roots, i.e.,
α = (-b + √b² – 4ac)/ 2a + and ß = (-b + √b² – 4ac)/ 2a

(ii) If b² – 4ac = 0; then it has two equal real roots, i.e.,
α = -b/2a and ß = -b/2a

(iii) If b² – 4ac < 0; then it has no real roots, i.e.,
α no real root and ß no real root 

Discriminant: b² – 4ac is called discriminant.
If D = b² – 4ac
Then, Discriminant = D
And we can write as follows:
α = (- b + √D)/2a + And ß = (-b – √D)/2a

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5. ARITHMETIC PROGRESSIONS

An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

This fixed number is called the common difference of the A.P. The common difference can be positive, negative or zero.

nth TERM OF AN A.P. :

Let a₁, a₂, a₃, ………………… be an A.P. whose first term a1 is a and the common difference is d.
Then,
The nth term a_n = a + (n –1) d

SUM OF FIRST n TERMS OF AN A.P.:
Let the first term of an A.P. be a, common difference be d and the sum of the first n terms be S.
A.P. = a, a + d, a + 2d, a + 3d …upto n terms.
The sum of the first n- terms of an A.P. is given by
S = n/2 [2a + (n-1)d]