Q1. Find the largest real root a of f(x) = x° −x−1=0 lying between 1 and 2. Perform three iterations by
- a) i)) bisection method (100 words)
- a) ii)) secant method (x₁ = 2, x₁ = 1). (100 words)
- b)) Find the number of positive and negative roots of the polynomial P(x) = x³-3x² +4x-5. Find P(2) and P'(2) using synthetic division method. (150 words)
- c)) Solve x³-9x+1=0 for the root lying between 2 and 4 by the method of false position. Perform two iterations. (150 words)
Key Points:
- Bisection method guarantees convergence by halving the root interval in each step.
- Secant method approximates the derivative using two previous points for faster convergence than bisection.
- Descartes' Rule of Signs predicts positive and negative real roots by counting sign changes in P(x) and P(-x).
- Synthetic division efficiently evaluates P(a) as the final remainder and P'(a) by re-applying to the quotient.
Answer: Numerical analysis provides various iterative methods to approximate the roots of non-linear equations. These methods are crucial when analytical solutions are not feasible. The given problem requires finding the largest real root of f(x) = x³ - x - 1 = 0 between 1 and 2, and then analyzing another polynomial. We will employ the Bisection, Secant, and False Position methods, along with Descartes' Rule of Signs and synthetic division. The Bisection method works by repeatedly halving the interval...