formed by subtracting each root from x and placing the product of the remainders equal to zero.

EXERCISE 136

1. Form an equation whose roots are (-1 ± √2).

Subtract each root from x, and write the product equal to zero.

186. The graphs of equations or functions of a higher degree than the second, may be plotted by the methods. already shown. This graphical method furnishes a convenient method of locating the real roots of higher equations.

In general, the number of real roots of an equation in x will be equal to the number of times the graph cuts the X axis. When the graph touches the X axis without crossing it, that is, when the graph is tangent to the X axis, equal roots are indicated. If the graph approaches the X axis and then recedes without touching the axis, the roots are imaginary.

Born at Fontenay-le-Comte, 1540. Died at Paris, 1603.

The greatest French algebraist of the sixteenth century.

He was a lawyer, who took up mathematics as a relaxation. His In Artem Analyticam Isagoge, 1591, is the first work with a symbolic treatment of algebra.

Vieta arrived at an incomplete understanding of the relations between the coefficients and roots of an equation.

He also wrote on geometry and trigonometry.

Let y

=

x3

=

0

3x2 + 4x − 2, then any value of x that makes y will be a root of the equation; hence any point of the graph that lies on the X axis indicates a root of the equation.

For any negative value of x, y is negative, hence the curve cannot touch the X axis on the left of the Y axis, and as for positive values of a greater than 2, the value of y becomes positively greater, the graph goes away from the X axis. Hence, there is only one real root, x = 1, the other two roots of this cubic equation are imaginary. (Fig. I.)

Hence the roots are at -3, between 0 and 1, and between 2 and 3.

Without solving, discuss the roots of the equation:

21. Determine the values of k that the equation (k2)x210kx+250 may have equal roots.

22. Form the quadratic equation, one of whose roots is 3 + √2, and the sum of whose roots is 6.

23. For what values of m are the roots of 2mx2 + 7mx x+5

5m equal?

=

24. Form the quadratic equation in which one root is 3+2 and the product of the roots is 7.

1

25. Locate the roots of the equation, x3-3x2 - 9x+

26. Form an equation whose roots shall be the cubes of the roots of the equation 2x(x - a) = a2.

27. Under what conditions will the equation x2 + px = q have roots that are reciprocals of each other?

28. Locate the roots of x3 + 4x2 — 7 = 0.