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Theory of Equation

30. Oct 2019
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Theory of Equation

1. Theory of Equations – 18UMTC12 Mrs.P.Kalaiselvi, M.Sc.,M.A., Ms.S.Swathi Sundari, M.Sc.,M.Phil.,
2. Preliminaries - Polynomials
3. Preliminaries Solving an equations by using various methods Algebraic Long Division Method Synthetic Division Method Completing The Square Method Factorization Method Special Quadratic Formula (only for solving quadratic equation)
4. Preliminaries – Solving an equation by using Algebraic Long Division Method
5. Preliminaries – Solving an equation by using
6. Preliminaries – Solving an equation by using Factorization Method
7. Preliminaries – Solving an equation by using
8. Preliminaries – Solving an equation by using Special Quadratic Formula
9. Preliminaries – Solving an equation by using Special Quadratic Formula
10. RESULTS Remainder Theorem : If f(x) is a polynomial, then f(a) is the remainder when f(x) is divided by x-a. If f(a) and f(b) are of different signs, then at least one root of the equation f(x)=0 must lie between a and b. If f(x)=0 is an equation of odd degree, it has at least one real root whose sign is opposite to that of the last term. If f(x)=0 is an equation of even degree and the absolute term is negative, equation has at least one positive root and at least one negative root. Every equation f(x)=0 of the degree has n roots and no more. In an equation with real (rational) coefficients, imaginary (irrational) roots occur in pairs.
11. Relation Between Roots and the Coefficients
12. Transformations of Equations Reciprocal Equations : An equation in which the reciprocal of every root of the equation is also its root is called a reciprocal equation. (Or) An equation, which remains unchanged when x is replaced by 1/x is called a reciprocal equation. Note: In such an equation, the coefficients from one end are equal to the coefficients from the other end (or) Equal in magnitude and opposite in sign. Remark:  When an odd degree equation, • If the coefficients have like signs, then -1 is a root. • If the coefficients of the terms equidistant from the first and last have opposite signs, then +1 is a root.  The degree is even and the coefficients of the terms equidistant from the first and last are equal and have the same sign.
13. Transformations of Equations  If are the roots of f(x)=0, the equation i. Whose roots are is f(1/x) =0. ii. Whose roots are is f(x/k) =0. iii.Whose roots are is f(x+h) =0. iv.Whose roots are is f(x-h) =0. v. Whose roots are is f(√x) =0.  In an reciprocal equation, increasing by h the roots of the equation is the same as diminishing the roots by –h.
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