The Theory of Equations; With an Introduction to the Theory of Binary Algebraic Forms Volume 25 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1881 edition. Excerpt: ...have been expected from the magnitude of the coefficients in f2 (1). In fact when the root of f2 (z) is substituted in f; (at) the positive part is nearly equal to the negative part. This is always an indication that two roots af the proposed equation are nearly equal. There are in the present instance two positive roots between 3 and 4. Subdividing the intervals, we find the two roots still to lie between 3-2 and 3'3; so that they are very close together. We see here another illustration of the continuity which exists between real and roots. If f3(z') turned out to be zero, the roots would be actually equal. If it turned out to be small negative number, the two nearly equal roots would be imaginary. 8. Analyse the equation The quadratic function is found to have imaginary roots. Ans. One real root between 0, 1; four imaginary roots. and the calculation may stop. Am. Two real roots; in the intervals--1,0, 5, 6. ll. Examine how the roots of the equation are situated in the several intervals between the numbers-0:, -7, 6, + no. Whenever, as in this example, any quantity makes one of the auxiliary functions vanish (here--7 satisfies f; (z) = 0), the zero may be disregarded in counting the number of changes of sign in the corresponding row; for, since the signs on each side of it are different, no alteration in the number of changes of sign in the row could take place, whatever sign be supposed attached to the vanishing quantity. The roots are all real. There is one root between--co and--7; and two be tween--7 and 6. 92. Conditions for the Reality of the Roots of an Equation.--The number of Sturm's functions, including f(: e), f '(a-), and the n--1 remainders, will in general be n + 1. In certain cases, owing to the absence of terms...