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Roger Morris
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1 THEORY OF EQUATIONS SYNOPSIS. Polyomil Fuctio: If,, re rel d is positive iteger, the f)x) = + x + x x is clled polyomil fuctio.. Degree of the Polyomil: The highest power of x for which the coefficiet is o-zero i polyomil fuctio, is clled the degree of the fuctio.. Costt Polyomil: If the degree of the polyomil fuctio is zero, the tht polyomil is clled costt fuctio. 4. Zero Polyomil: If the coefficiets of polyomil re ll zeros, the tht polyomil is clled zero polyomil.zero polyomil hs o degree. The domi of zero polyomil is R. The rge is subset of R 5. Polyomil Equtio: If,,,.. re rel d is positive iteger the f(x) = + x + x + + x = is clled polyomil equtio i x, with rel coefficiets. If, the f(x) = is equtio of degree. 6. Lier, Qudrtic, Cubic, Biqudrtic equtios: polyomil equtios of degree,,,4 re respectivelyclled s lier, qudrtic, cubic, biqudrtic equtios. 7. Divisio Algorithm: Let f(x), g(x) be the polyomils of degree d m respectively such tht m <. The there exist two polyomils q(x) d r(x) uiquely such tht f(x) = q(x). g(x) + r(x). The degree of q(x) is ( m). 8. Remider Theorem: If polyomil f(x) is divided by (x ) the the remider is f().

2 9. Fudmetl Theorem: Every polyomil equtio of degree hs tlest oe root rel or imgiry.it hs oly roots rel or imgiry d o more.. Every th degree equtio hs exctly roots rel or imgiry.. If α, α,α...α re the roots of x + x + x , the S =, S =, S =,..., S = ( ) =. Where S stds for the sum of the products of the roots tke r t time.. For the equtio x + p x + p x p = i) α = p p ii) α = + pp p p iii) α 4 4 = p 4p p + p + 4pp 4p4 iv) α β = p p p v) α βγ = pp 4p4 Note: For the equtio x + px + px + p = α β = p pp 4. For the equtio x + p x + p x p, = i) S r + Sr p + Sr p S pr + rpr =, if r d ii) Sr + Sr p + Sr p Sr p =, if r >. Where S r deotes the sum of the r th powers of the roots of the equtio.

3 5. The sum of the r th powers of the roots of the equtio x + p x - +p x -...+p - x +p = is the coefficiet of x -r i the expsio of ( x) ( x) xf i descedig powers of x. f 6. To remove the secod term from th degree equtio, the roots must be dimiished by d the resultt equtio will ot coti the term with x If α, α,...α re the roots of f(x) =, the equtio i) Whose roots re,... is f =. α α α x x ii) Whose roots re kα, kα,...,kα is f =. k iii) Whose roots re α h, α h,... α h is f(x+h) =. iv) Whose roots re α + h, α + h,...α + h is f(x-h) =. v) Whose roots re α, α... α is ( y) f = 8. If α, β re the roots of the qudrtic equtio f(x) =, the the equtio whose roots re α + b, β+b is x b f =. 9. A equtio i which the reciprocl of every root of the equtio is lso its root is clled reciprocl equtio. I such equtio, the coefficiets from oe ed re equl to coefficiets from the other ed (or) Equl i mgitude d opposite i sig.. I y equtio with rtiol coefficiets, irrtiol roots occur i cojugte pirs.

4 . I y equtio with rel coefficiets, complex roots occur i cojugte pirs.. If α is r - multiple root of f(x) =, the α is (r-) - multiple root of f (x)= d (r-) - multiple root of f (x) = d o multiple root of f r- (x) =.. i) If f(x) = x + p x p - x + p d f() d f(b) re of opposite sig, the tlest oe rel root of f(x) = lies betwee d b. 4. Descrte's rule of sigs If f(x) is polyomil (with rel coefficiets with the terms rrged i descedig powers of x), the umber of rel positive roots of f(x) = does ot exceed the umber of chges i sigs of the coefficiets of f(x), d the umber of egtive rel roots of f(x) = does ot exceed the umber of chges of sigs of f(-x). 5. i) The equtio of lowest degree with rtiol coefficiets hvig root + i b is x 4 ( b)x +(+b) =. ii) The equtio of lowest degree with rtiol coefficiets hvig root + b is x 4 (+ b)x +(-b) =. 6. The coditio tht the roots of x + bx +cx+d= my be i A.P. is b +7 d = 9bc. 7. The coditio tht the roots of x + bx + cx + d = my be i G.P. is c = b d. 8. The coditio tht the roots of x +bx +cx + d = my be i H.P. is c +7d = 9bcd. 9. i) If the reciprocl of every root of equtio is lso root of it, the the equtio is sid to be reciprocl equtio.

5 ii)if the coefficiets from oe ed of equtio re equl i mgitude d sig to the coefficiets from the other ed, the the equtio is sid to be the reciprocl equtio of First Type. iii) If the coefficiets from oe ed of equtio re equl i mgitude d opposite i sig to the coefficiets from the other ed, the the equtio is sid to be the reciprocl equtio of Secod Type.. i) x = - is root of the reciprocl equtio of first type d of odd degree ii) x = is root of the reciprocl equtio of secod type d of odd degree. iii) x = ± re two roots of reciprocl equtio of secod type d of eve degree.

ALGEBRA. Set of Equations. have no solution 1 b1. Dependent system has infinitely many solutions

ALGEBRA. Set of Equations. have no solution 1 b1. Dependent system has infinitely many solutions Qudrtic Equtios ALGEBRA Remider theorem: If f() is divided b( ), the remider is f(). Fctor theorem: If ( ) is fctor of f(), the f() = 0. Ivolutio d Evlutio ( + b) = + b + b ( b) = + b b ( + b) 3 = 3 +