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Estimating the Roots of a Polynomial
brainstorm
What do we already know that can help us find the roots of a polynomial?
Board of Studies
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Estimating the Roots of a PolynomialIt is not always possible to find the roots of a polynomial equation by algebraic methods. Use of stationary points can help...except we haven't covered that topic yet!
There are two iterative methods of estimating roots of polynomial equations in this course. These are called halving the interval and Newton’s method of approximation. In these methods, we guess a solution and then use a process over and over to produce closer approximations.
These methods can also be used to find roots of functions that are not polynomials, as long as the function is continuous over the interval.
presspress
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Estimating the Roots of a PolynomialIf f (x) is continuous for a ≤ x ≤ b and f (a) and f (b) have oppositesigns, then there is at least one root of f (x) = 0 in that interval.
We can find an approximation to the root between a and b by halving this interval.
If we halve the interval several times, the approximation to the root will usually, but not always, become more accurate.
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Estimating the Roots of a PolynomialIf f (a) < 0 and f (b) > 0, then there are three possibilities:1. f (a+b) = 0 means a+b is a root of the equation 2 2
2. f(a+b) < 0 means that the root lies between x = a+b and x=b2 2
3. f(a+b) > 0 means the root lies between x=a and x=a+b2 2
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Groves Ex 9.1
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Estimating the Roots of a PolynomialNewton’s method is a different way of approximating the root of a polynomial equation. It generally gives a more accurate approximation than the method of halving the interval, as well as taking fewer steps to get this approximation.Sketching y = f (x), a continuous function, shows how Newton’s method works.
If x = a is close to the root of the equation f (x) = 0, then the xintercept (a1) of the tangent at a is usually closer to the root.
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Estimating the Roots of a Polynomial
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Groves Ex 9.2
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Jones and Couchman
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Fitzpatrick