Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a 2 + b 2 = c 2.

Learn how to find the distance between two points by using the distance formula, which is an application of the Pythagorean theorem. We can rewrite the Pythagorean theorem as d=√((x_2-x_1)²+(y_2-y_1)²) to find the distance between any two points. Created by Sal Khan and CK-12 Foundation.

The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae , it is one of the basic relations between the sine and cosine functions.

Question Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two sides are divided in the same ratio. Solution In ∆ A B C, line D E parallel to B C and intersects A B at D and A C at E. We have to prove that, D E divides the two sides in the same ratio i.e.,. A D D B = A E E C

.the Chinese remainder theorem can be generalized to ideals. is true. Here is what I imagine: Say the residue classes mod 3 are [0] = {., − 6, − 3, 0, 3, 6,. } [1] = {., − 5, − 2, 1, 4, 7,. } [2] = {., − 4, − 1, 2, 5, 8,. }

The Chinese Remainder Theorem Suppose we wish to solve x = 2 ( mod 5) x = 3 ( mod 7) for x. If we have a solution y, then y + 35 is also a solution. So we only need to look for solutions modulo 35. By brute force, we find the only solution is x = 17 ( mod 35).

A similar argument provides a proof of the Chinese Remainder Theorem. Theorem 1.7.2: Chinese Remainder Theorem If m and n are relatively prime, and 0 ≤ a < m and 0 ≤ b < n, then there is an integer x such that x mod m = a and x mod n = b. Proof More general versions of the Pigeonhole Principle can be proved by essentially the same method.

Then the Taylor series. ∞ ∑ n = 0f ( n) (a) n! (x − a)n. converges to f(x) for all x in I if and only if. lim n → ∞Rn(x) = 0. for all x in I. With this theorem, we can prove that a Taylor series for f at a converges to f if we can prove that the remainder Rn(x) → 0. To prove that Rn(x) → 0, we typically use the bound.

Remainder theorem, factor theorem and their uses are the key concepts covered in this chapter. For further assistance, students can get Selina Solutions for Class 10 Mathematics Chapter 8 Remainder and Factor theorems PDF, from the links provided below. Selina Solutions Concise Maths Class 10 Chapter 8 Remainder and Factor Theorems Download PDF

NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem From equation 1, 2 and 3 we get 10. Find the expansion of (3x2 – 2ax + 3a2)3 using binomial theorem. Solution: We know that (a + b)3 = a3 + 3a2b + 3ab2 + b3 Putting a = 3x2 & b = -a (2x-3a), we get [3x2 + (-a (2x-3a))]3 = (3x2)3+3(3x2)2(-a (2x-3a)) + 3(3x2) (-a (2x-3a))2 + (-a (2x.

BINOMIAL THEOREM 8.1 Overview: 8.1.1An expression consisting of two terms, connected by + or – sign is called a 1 4 binomial expression. For example, x + a, 2x – 3y, − , 7x−, etc., are all binomial 3x5yexpressions. 8.1.2 Binomial theorem If a and b are real numbers and n is a positive integer, then + b)n =nC an 1 – 2 0+ nC an b1 + nC an b2 +.

Pythagoras, (born c. 570 bce, Samos, Ionia [Greece]—died c. 500–490 bce, Metapontum, Lucanium [Italy]), Greek philosopher, mathematician, and founder of the Pythagorean brotherhood that, although religious in nature, formulated principles that influenced the thought of Plato and Aristotle and contributed to the development of mathematics and Wes.