By H. T. Clifford

ISBN-10: 0121767507

ISBN-13: 9780121767501

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Additional resources for An Introduction to Numerical Classification

Example text

As a superior limit and -fa as an inferior limit of the positive values of y. and 20, Hence the negative roots of the given equation lie between We f ^ and x, * all the roots between 24 and lie 20. 5522-... Ex. 2. -20xBy Between what 16 = 38 and 3. do the real roots of x6 -f 6 x4 + 16 x2 x8 lie? 41, the roots lie the roots lie between Ex. limits of y and Between what (1) 2) V 3) 6. between 21 and -21. By The 4, roots are 2, 2, 1 41, V -^3) . limits are the real roots of - aP - 10 x - 12 = 0, + 3 x*-3x + 3x- =0, x 5 - 11 x4 + 17 x' + 17 x' - 11 x + 4 a- 39 and ( 4 jr 1 2 1 - ?

Thus, = = G the vanishing of the discriminant indicates equal roots. TJiree Equal Roots. //= and G = 0. of V Ex. 1. different Ex. Equation I has three equal roots when = V Since all the roots 0. reduces to 2 s For, are zero, all the roots of I must be equal to one another. 2. Prove that equation from zero. If V in 35 cannot have three equal roots two roots in V are equal to each drawn about the roots of I ? other, but not zero, what inference can be Ex. 3. Compute the discriminant of X s Find the discriminant of 4 inference can be drawn from its value ?

Do the real roots of x6 -f 6 x4 + 16 x2 x8 lie? 41, the roots lie the roots lie between Ex. limits of y and Between what (1) 2) V 3) 6. between 21 and -21. By The 4, roots are 2, 2, 1 41, V -^3) . limits are the real roots of - aP - 10 x - 12 = 0, + 3 x*-3x + 3x- =0, x 5 - 11 x4 + 17 x' + 17 x' - 11 x + 4 a- 39 and ( 4 jr 1 2 1 - ? b, when an numodd f(x) contrary siyns, ber of roots of the equation f(x) = must lie between a and b ; if they give to f(x) the same ir/n either no root or an even number of roots must He between a and b.