Theory of Equations

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CHAPTER I - THEORY OF EQUATIONS DEFINITION 1 A function defined by n n-1 f ( x ) = p 0 x + p 1 x + ... + p n-1 x + p n where p 0 ≠ 0, n is a positive integer or zero and p i ( i = 0,1,2...,n) are fixed complex numbers, is called a polynomial function of degree n in the indeterminate x . A complex number a is called a zero of a polynomial f ( x ) if f (a ) = 0. Theorem 1 – Fundamental Theorem of Algebra Every polynomial function of degree greater than or equal to 1 has atleast one zero. Definition 2 n n-1 Let f ( x ) = p 0 x + p 1 x + ... + p n-1 x + p n where p 0 ≠ 0, n is a positive integer . Then f ( x ) = 0 is a polynomial equation of nth degree. Definition 3 A complex number a is called a root ( solution) of a polynomial equation f (x) = 0 if f ( a) = 0. THEOREM 2 – DIVISION ALGORITHM FOR POLYNOMIAL FUNCTIONS If f(x) and g( x ) are two polynomial functions with degree of g(x) is greater than or equal to 1 , then there are unique polynomials q ( x) and r (x ) , called respectively quotient and reminder , such that f (x ) = g (x) q(x) + r (x) with the degree of r (x) less than the degree of g (x). Theorem 3- Reminder Theorem If f (x) is a polynomial , then f(a ) is the remainder when f(x) is divided by x – a . Theorem 4 Every polynomial equation of degree n ≥ 1 has exactly n roots. Theorem 5 In a polynomial equation with real coefficients , imaginary roots occur in conjugate pairs. Corollary 1 If f (a) = 0 , the polynomial has a factor x – a. Corollary 2 – Factor Theorem A number a is a zero of the polynomial f(x) if and only if x – a is a factor of f (x) . 1. Two roots of the equation x 4 – 6 x 3 + 18 x 2 – 30x + 25 = 0 are of the form a + i b and b + i a . find all roots of the equation. 2.Two roots of the equation x 4 - 6 x 3 + a x 2 + b x + 25 = 0 are of the form p + i q and q + i p . Find the values of a and b and also the roots of equation. Theorem -6 In an equation with rational coefficients quadratic surd roots occur in pairs. That is , if a + √ b is a root of f (x) =0 , then a - √ b is also a root of the equation. 1. Solve the equation x 5 – x 4 + 8 x 2 – 9 x – 15 = 0, one root being -√ 3 and another 1 + 2√ -1 . 2.Find all the roots of the quartic equation x 4 +x 3 – x 2 – 2x -2 = 0, given that one of its roots is √ 2 . 3. Find the equation with rational coefficients, one of whose roots is √5 +√ 2. .

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Unit 3 Cubic and Biquadratic Equations
UNIT 3 CUBIC AND BIQUADRATIC EQUATIONS Structure 3.1 Introduction Objectives 3.2 Let Us Recall Linear Equations Quadratic Equations 3.3 Cubic Equations Cardano's Solution Roots And Their Relation With Coefficients 3.4 Biquadratic Equations Ferrari's Solution Descartes' Solution Roots And Their Relation With Coeftlcients 3.5 Summary - 3.1 INTRODUCTION In this unit we will look at an aspect of algebra that has exercised the minds of several ~nntllematicians through the ages. We are talking about the solution of polynomial equations over It. The ancient Hindu, Arabic and Babylonian mathematicians had discovered methods ni solving linear and quadratic equations. The ancient Babylonians and Greeks had also discovered methods of solviilg some cubic equations. But, as we have said in Unit 2, they had llol thought of complex iiumbers. So, for them, a lot of quadratic and cubic equations had no solutioit$. hl the 16th century various Italian mathematicians were looking into the geometrical prob- lein of trisectiiig an angle by straight edge and compass. In the process they discovered a inethod for solviilg the general cubic equation. This method was divulged by Girolanlo Cardano, and hence, is named after him. This is the same Cardano who was the first to iiilroduce coinplex numbers into algebra. Cardano also publicised a method developed by his contemporary, Ferrari, for solving quartic equations. Lake, in the 17th century, the French mathematician Descartes developed another method for solving 4th degree equrtioiis. In this unit we will acquaint you with the solutions due to Cardano, Ferrari and Descartes. Rut first we will quickly cover methods for solving linear and quadratic equations.
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