Class 9 Math, Euclid's Geometry Dear Class 9 students and fellow educators, 'Jsunil Sir' welcomes you to an enlightening journey into the world of Euclid's Geometry at JSUNIL TUTORIAL. This comprehensive guide offers a clear and easy-to-understand introduction to the principles laid down by the ancient mathematician Euclid. Whether you're a budding learner or a dedicated teacher, this resource is designed to deepen your understanding of geometry. Explore the foundations, sharpen your math skills, and lay a strong groundwork for academic success. Dive into Euclid's Geometry with us today EUCLID’S Postulates
1. A straight line may be drawn from any point to any other point. 2. A terminated line (line segment) can be produced indefinitely. 3. A circle may be described with any centre and any radius. 4. All right angles are equal to one another. 5. If a straight line falling on two straight lines makes the interior angles on the same side of it, taken together less than two right angles, then the the two straight lines if produced indefinitely, meet on that side on which the sum of angles is taken together less than two right angles. Euclid used the term postulate for the assumptions that were specific to geometry and otherwise called axioms. A theorem is a mathematical statement whose truth has been logically established. Visit Page Link to download solved Questions and Guess test Paper CBSE Class IX Introduction to Euclid's geometry Proof of this factor theorem
Let p(x) be a polynomial of degree greater than or equal to one and a be areal number such that p(a) = 0. Then, we have to show that (x – a) is a factor of p(x).Let q(x) be the quotient when p(x) is divided by (x – a). By remainder theorem Dividend = Divisor x Quotient + Remainder p(x) = (x – a) x q(x) + p(a) [Remainder theorem] ⇒ p(x) = (x – a) x q(x) [p(a) = 0]⇒ (x – a) is a factor of p(x) Conversely, let (x – a) be a factor of p(x). Then we have to prove that p(a) = 0 Now, (x – a) is a factor of p(x)⇒ p(x), when divided by (x – a) gives remainder zero. But, by the remainder theorem, p(x) when divided by (x – a) gives the remainder equal to p(a). ∴ p(a) = 0 Proof of remainder theorem. Let q(x) be the quotient and r(x) be the remainder obtained when the polynomial p(x) is divided by (x–a). Then, p(x) = (x–a) q(x) + r(x), where r(x) = 0 or some constant. Let r(x) = c, where c is some constant. Then p (x) = (x–a) q(x) + c Putting x = a in p(x) = (x–a) q(x) + c, we getp(a) = (a–a) q(a) + c ⇒ p(a) = 0 x q(a) + c ⇒ p(a) = c This shows that the remainder is p(a) when p(x) is divided by (x–a).1. [a is –8.] Download study material for polynomial |
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