Mathematics Formula for Class 10

Euclid's division lemma
For any two positive integers a and b, there exist (unique) whole numbers q and r such that
a = bq + r, 0≤ r <b.
Euclid's division algorithm is a process based on Euclid's division lemma. According to this, the HCF of two positive integers a and b, with a > b, is obtained as follows:
Step 1: Apply Euclid's division lemma on a and b, a = bq + r, 0 ≤ r <b for some whole numbers q and r.
Step 2: If r = 0, then HCF (a, b) = b.
If r≠0, apply Euclid's division on b and r.
b = rq₁ +r₁, 0≤ r₁<b for some whole numbers q₁ and r₁
Step 3: If r₁ = 0, then HCF (b,r) = r.
But HCF (a, b) = HCF (b,r) ⇒ HCF (a, b) = r.
Continue the above process until the remainder is zero.
Last non-zero remainder = HCF (a, b)

Fundamental Theorem of Arithmetic
Every composite positive integer n can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
HCF and LCM of two (or more) natural numbers can be obtained by using Fundamental Theorem of Arithmetic. This method is also called prime factorisation method.
Relationship between HCF and LCM of two natural numbers
HCF × LCM = product of two natural numbers.
LCM of two natural numbers is always divisible by their HCF.

Theorem 1: If a is any natural number and p is a prime number such that p divides a², then p divides a.
Generalisation of the theorem is:
If a, n are any natural number and p is a prime number such that p divides aⁿ, then p divides a.
Theorem 2: If a and b are any natural numbers and p is a prime number such that p divides ab then p divides a or p divides b or p divides both.

Rational and Irrational numbers
A number that can be expressed in the form of p/q, where p, q are integers, q≠0, and p, q are co-prime is called a rational number.
A number that cannot be expressed in the above form is called an irrational number.

Theorem 3: If x is a rational number whose decimal expansion terminates, then x can be expressed as x = p/q where p, q are integers, q≠0 and p, q are co-prime, and the prime factorisation of q is of the form 2^m5ⁿ where m, n are non-negative integers.

Theorem 4: If x = p/q where p, q are integers, q≠0 and p, q are co-prime such that the prime factorisation of q is of the form 2^m5ⁿ where m, n are non-negative integers, then x has decimal expansion which terminates.

Theorem 5: If x = p/q where p, q are integers, 90 and p, q are co-prime such that the prime factorisation of q is not of the form 2^m5ⁿ where m, n are non-negative integers, then x has a decimal expansion which is non-terminating repeating (recurring).
Thus, if x is a rational number and x = p/q where p, q are integers, q≠0 and p, q are co-prime such that the prime factorisation of q has a prime factor other than 2 or 5, then the decimal expansion of x is non-terminating repeating.
The decimal expansion of every rational number is either terminating or non-terminating repeating and conversely every decimal number either with terminating decimal expansion or with non-terminating repeating decimal expansion is a rational number.

Value of a Polynomial:If p(x) is a polynomial and ‘a’ is any real number, then the real number obtained by replacing x by ‘a’ in p(x), is called the value of p(x) at x = a and is represented by p(a).
For example, Let p(x) ≡ x² + 2x + 5 be a polynomial. The value of p(x) at 3 denoted by p(3) is given by
p(3) = 3² +  2 × 3 + 5 = 20.

Relations between zeroes and coefficients of a polynomial
(i) If a and b are zeroes of the quadratic polynomial ax² + bx + c, then
α+ β = -b/a and αβ = c/a
(ii) If α, β and γ are zeroes of the cubic polynomial ax³ + bx² + cx + d, then
α+β+γ = -b/a
αβ + βγ + γα = c/a and αβγ = -d/a.

Division Algorithm for polynomials
If f(x) and g(x) are any two polynomials with g(x)≠0, then there exist (unique) polynomials q(x) and r(x) such that
f(x) = g(x) q(x) + r(x) where,
r(x) = 0 or deg r(x) < deg g(x).
If deg p(x) ≥ deg g(x), then deg q(x) = deg f(x) – deg g(x).

Graph of a Polynomial
The graph of a polynomial p(x) is a freehanded curve passing through points (x₁, y₁), (x₂, y₂), (x₃, y₃) ......... etc., where y₁, y₂, y₃ ....... are the values of polynomial p(x) at x₁, x₂, x₃ ....... respectively.
Example: Graph of p(x) ≡ 2x +1

Geometrical meaning of the zeros of a Polynomial
The zeros of a polynomial are the x-coordinates of the points where the graph of polynomial intersects or touches x-axis.

Example:
In the following polynomial p(x), 1, 4, 7 are zeros of p(x).

• If ax² + bx + c is factorizable into a product of two linear factors, then the roots of the quadratic equation ax² + bx + c = 0 can be found by equating each factors to zero.
For example:
x²– 5x + 6 = 0
⇒ x – 3x – 2x + 6 = 0
⇒ x(x – 3) – 2 (x – 3) = 0
⇒ (x – 3) (x – 2) = 0
⇒ x – 3 = 0 or x – 2 = 0
Therefore, x = 3 or x = 2
⇒ x = 3, 2 are roots of x²– 5x + 6 = 0

Sridharacharya’s Formula or Discriminant Rule

The following formula used to find the roots a, b of quadratic equation ax² + bx + c = 0, a ≠ 0, is given by,

Selection of term in an AP
In some particular problems we require certain number of terms in AP. For convenience we adopt following pattern

Number of terms	Terms	Common difference
3	a – d, a, a + d	d
4	a – 3d, a – d, a + d, a + 3d	2d
5	a – 2d, a – d, a, a + d, a + 2d	d
6	a – 5d, a – 3d, a – d, a + d, a + 3d, a + 5d	2d

A sequence a₁, a₂, a₃.......is an AP, if a_n+1– a_n is independent of n.
For example:
2, 5, 8, 11, .......is an AP because
a_n+1– a_n = {2+(n + 1 – 1)3} – {2 + (n – 1)3}
= {2 + 3n – 2 – 3n + 3} = 3 (independent of n)

• A sequence a₁, a₂, a₃.........., a_n, is an AP, if and only if n^th term an is a linear expansion of n and the co-efficient of n is common difference of the AP.
For example:
If ‘a’ be the first term and ‘d’ the common difference then
a_n = a + (n – 1) d
= a + nd – d = dn + (a – d)

Bisector Theorems
• The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.
For example:

If ABC is a triangle and AD is the bisector of ∠A, then AB/AC=BD/CD

• The external bisector of an angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle.

For example:
If ABC be a triangle and AD is the external bisector of ∠A, then AB/AC = BD/CD