Synthetic division

The cubic equation can be factored. You get to use some more powerful algebra in the form of the rational root theorem and synthetic division to find the factors. (These topics are covered thoroughly in Algebra II For Dummies. )... 

Clearly, any polynomial in the denominator can be expanded into an ascending series by synthetic division in the present case, we can use the binomial theorem written generally as... 

The most effective way of finding the roots of nonlinear equations is to devise iterative algorithms that start at an initial estimate of a root and converge to the exact value of the desired root in a finite number of. steps. Once a root is located, it may be removed by synthetic division if the equation is of the polynomial form. Otherwise, convergence on the same root may be avoided by initiating the search for subsequent roots in different region of the feasible space. 

If the nonlinear equation being solved is of the polynomial form, each real root (located by one of the methods already discussed) can be removed from the polynomial by synthetic division, thus reducing the degree of the polynomial to (n - 1). Each successive application of the synthetic division algorithm will reduce the degree of the polynomial further, until all real roots have been located. 

A simple computational algorithm for synthetic division has been given by Lapidus [4], Consider the fourth-degree polynomial... 

In general notation, for a polynomial of nth-degree, the new coefficients after application of synthetic division are given by... 

Example 13 Solution of nth-Degree Polynomials and Transfer Functions Using the Newton>Raphson Method with Synthetic Division and Eigenvalue Method. Consider the isothermal continuous stirred tank reactor (CSTR) shown in Fig. El.3. 

Using the Newton-Raphson method with synthetic division or eigenvalue method, determine the roots of the transfer function for a range of values of the proportional gain K. and calculate the critical value of. S above which the system becomes unstable. Write the program so that it can be used to solve nth-degree polynomials or transfer functions of the type shown in the above equation. 

Method of Solution In the Newton-Raphson method with synthetic division, Eq. (1.42) is used for evaluation of each root. Eqs. (1.53)-(1.55) are then applied to perform synthetic division in order to extract each root from the polynomial and reduce the latter by one degree. When the nth-degree polynomial has been reduced to a quadratic... 

In order to determine whether the system is stable or unstable, the two polynomials are combined, as shown in the Method of Solution, using as the multiplier of the polynomial from the numerator of the transfer function. Function NRsdivision (which uses the Newton-Raphson method with synthetic division algorithm) or function roots (which uses the eigenvalue algorithm) is called to calculate the roots of the overall polynomial function and the sign of all roots is checked for positive real parts. A flag named stbl indicates that the system is stable (all negative roots stbl = 1) or unstable (positive root stbl = 0). 

The polynomial may have no more than a pair of complex roots, A root of nth-degree polynomial is determined by Newton-Raphson method. This root is then extracted from the polynomial by synthetic division. This procedure continues until the polynomial reduces to a quadratic. 

Solves nth-degree polynomials and transfer functions using the Newton-Raphson method with synthetic division NRsdivision.m). 

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