By Cajori F.

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19. proofs of this theorem, the fourth (1849) being a simplification of the first It is (1799). The one given here is in substance Gauss's proof of 1849. geometrical in character, and is open to the objection raised in the foot-note . of 22, , ; SOME ELEMENTARY 1'UOPEUTIES OF EQI/ATIONS "27 polynomial f(z) of the wtli degree has real coefficients only. We wish to prove that there exists always at least one value of z, either real or complex, which causes the polynomial J\z) to vanish. Let = x -f iy, then, by z 22, the variable' represents points in a plane, and the function f(z) has a definite value at each 8, we may write J\z) = I'+iQ, point in the plane.

And /"(a-) (a;) = gives x /"( 3) is maximum. Ex. 2. - 36 + sr = - 2, negative. Find the 75. or = = + -JO x + 12 a + 30. (> a* - 3. We Hence /( 8 + is /" (- 2) 2 15 x2 -f is 36 z + o. r) values of f(x) = 3) 2 y^ + and is a 3 r2 LOCATION OF THE ROOTS OF AN EQUATION b Between, two xnwwive real roots a an<\ 45. Rolle's Theorem. r) = there lien at leant one real root of the ()' equation f'(x) = 0. Let the curve in this figure be the graph of f(x) = 0. *) 0. the curve bends down and -2V N then up.

Remove the second term of 4 x 33. Removal of Second Term mation of the general cubic &00* +3M jr ^ 4 + 8 x3 4- must be increased by cubic being - -* UQ ing, we obtain Put // x 4- + in the Cubic. 'J 7v; + 1>;\ = 0. 1 12 = 0. In the transfor- = into another, deprived of the second term, root r2 4 -f lO,* -f we notice that each \ the sum of the roots in the given =x -f- , f> n then x =y -- UQ Substitut- ELEMENTARY TRANSFORMATION OF EQUATIONS Expanding, and collecting the powers of y, we where coefficients 37 the different of get # = bb, 6 b?

### Theory of Equations by Cajori F.

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