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483 Rh 614. In giving values to x it is evident that the substitution of a root gives y=0, that is, the ordinate, or distance from the Axis of Abscissas, is 0; hence where the graph cuts the Axis of X we have the location of a real root, and the graph will cross the Axis of X as many times as the equation has real and unequal roots. If the roots of the equation be imaginary, the curve will not touch the Axis of X.

EXAMPLES XLVIII. h.

Construct the graphs of the following functions :

1. 24-5. 3. 22-5. 5. «8-2. 7 wt! 30? +3. 2 42741. 4 w4a-l. 6. 28-5443. 8. #3434245 a-12.

SOLUTION OF HIGHER NUMERICAL EQUATIONS. Commensurable Roots.

615. A real root which is either an integer or a fraction is said to be commensurable.

By Art. 582 we can transform an equation with fractional coefficients into another which has all of its coefficients integers, that of the first term being unity: hence we need consider only equations of this form. Such equations cannot have for a root a rational fraction in its lowest terms [Art. 575], therefore we have only to find the integral roots. By Art. 570 the last term of f(x) is divisible by every integral root, therefore to find the commensurable roots of f(x) it is only necessary to find the integral divisors of the last term and determine by trial which of them are roots.

616. Newton’s Method. If the divisors are small numbers we may readily ascertain by actual substitution whether they are roots. In other cases we may use the method of Arts. 562 and 566 or the Method of Divisors, sometimes called Newton’s Method.

Suppose a to be an integral root of the equation

= ye pea 4 ot Dae +p, = 0.