If f(x' + c)

=

box'n + b1x'n−1 +

x'

(9)

+ bn_2x12 + bn-12′ + b2 be divided by x', as indicated on the right of (9), the remainder is b, and the quotient is

If the quotient q1(x') is divided by x', the remainder is b2-1 and the quotient

Continuing this process it is seen that the successive remainders are the coefficients of (6) in the order bn, bn−1, bn-2, bo. But if the indicated

.

divisions be performed on both sides of (9), if the respective quotients thus obtained be divided by x c and x', and so on, the successive remainders obtained on the left side of (9) will equal those obtained on the right, namely, bn, bn-1, bn-2, I bo. Hence we have the

Theorem. The coefficients of f(x' + c) may be obtained by dividing f(x) by xc, the quotient by xc, etc.

1. Construct the graph of each of the functions below, and find the function having the same graph referred to a new y-axis c units to the right. Verify the result by direct substitution of x' + c for x.

=

2. Show that the second term of the function y 23 be removed if the y-axis is translated 2 units to the right.

6x2+7x+4 will

Deduce a rule for removing the second term of ax3 + bx2 + cx + d. 3. Determine a translation of the y-axis so that the graph of x3 + 3x2 4 in the old system will be the graph of x3- 3x + 2 in the new system. Plot the graphs of x3 and 3x 2 on the same axes and from them determine approximately the roots of the equation x3 – 3x + 2 = 0, and hence of the equation x3 + 3x2 - 4 = 0.

4. Plot the graph of x4 - 2x3 3x2+4x+2 and determine the coordinates of the maximum and minimum points and of the points of inflection. Find the function which has the same graph referred to a y-axis 1 unit to the left.

53. Horner's Method of Solution of Equations. This method enables us to compute irrational roots as accurately as may be desired.

EXAMPLE. Find, correct to two decimal places, the real root of the equation

f(x) = x3 + x − 47 = 0.

(1)

Plotting the graph of f(x) we get the curve in the figure, which shows that there is a real root between 3 and 4. As the coefficients of f(x) are integers, and that of x3 is unity, this root is not fractional, and it therefore must be irrational.

Now move the y-axis 3 units to the right.

บ ใบ

-241

-18

FIG. 72.

9

The new equation, omitting the primes on the x's, is

No confusion should arise from omitting the primes

if it be remembered that the graph of the polynomial

in (2) is the curve in the figure referred to the axes O'X and O'Y'. Equation (2) has a root between 0 and 1, which from the graph appears to be about 0.5.

Dividing (2) by x

0.5, we have

As the remainder is negative, the graph lies below the x-axis at x = and hence, from the figure, the root is larger than 0.5. Try x = 0.6.

= 0.5

The remainder is now positive and hence the graph lies above the x-axis

at x 0.6. Therefore the root of (2) lies between 0.5 and 0.6.

=

The graph of the polynomial on the left is the same curve referred to the axes O"X and O"Y", and it shows that equation (3) has a root between O and 0.1. As the square and cube of a number less than 0.1 are very small, an approximate value of the root may be obtained by neglecting r3 and x2 in (3) and solving the resulting linear equation

As the remainder is positive, the graph shows that the root is less than 0.02. Try 0.01.

This remainder is negative, and hence the graph shows that the root is greater than 0.01. Hence the root of (3) lies between 0.01 and 0.02.

Then the root of equation (2) lies between 0.51 and 0.52, and that of (1) between 3.51 and 3.52. Hence the real root of (1), correct to two decimal places, is x

=

3.51.

Which is the closer approximation to the root, 3.51 or 3.52? Why?

EXERCISES

Find all the real roots of the equations following, obtaining irrational roots to two decimal places.

NOTE. To find negative roots by Horner's method, replace x in the equation by - x. The graph of ƒ(− x) is symmetrical to that of f(x) with respect to the y-axis, and the roots of the equation f(x) = 0 will be equal numerically to those of f(x) = 0, but have opposite signs. Hence the negative roots of an equation f(x) = 0 may be found by finding the positive roots of ƒ( − x) = 0, and changing their signs.

B,

NOTE. If an equation has both rational and irrational roots, it is advisable to find the rational roots first. Suppose they are a, B, Y, etc. Then divide the equation by xa, the resulting equation by x-ẞ, the new equation by xy, etc., thus obtaining a simpler equation whose irrational roots are the same as those of the given equation, and then solve this simpler equation by Horner's method.

depth) rests upon

20. A cast iron rectangular girder (breadth = supports 12 feet apart and carries a weight of 2000 pounds at the center. In order that the intensity of the stress may nowhere exceed 4,000 pounds per square inch, it is determined that the depth d of the girder in inches must satisfy the equation 80ď3 – 81d2 – 17,280 = 0. Find d and the crosssectional area.

21. The depth of flotation of a buoy in the form of a sphere is given by the equation x3 3rx2 + 4r3s = 0, where r is the radius and s is the specific gravity of the material. What is the depth for such a buoy whose radius is 1 foot and specific gravity is 0.786?

22. The cross section of the retaining wall of a reservoir is designed as indicated in Fig. 73. The allowable height x of the upper portion is given by the equation x3 + 32x2 – 69x – 88 = 0, where x is expressed in terms of a unit of 10 feet. Find the allowable height to three significant figures.

=

23. The allowable height of the lower portion of the wall in Exercise 22 is given by the equation y1 – 2.7y3 + 18.4y2 – 123y – 1.46 0, where y is in terms of a unit of 10 feet. Find the height of the lower portion to three significant figures.

E

D C

24. In Exercise 22 the slope of the water front BC to the vertical is, of DE is %, of AE is 70% The width of the top is 6 feet. What must be the width of the lower portion?

25. The load P, concentrated at the center, which a homogeneous elliptical plate can support is given by the formula

P

where h is the thickness of the plate = in., R is the maximum safe unit stress for the material 16,000 pounds per square inch, m is the ratio of the breadth to the length of the ellipse.

of m.

=

=

600π, and Find the value

26. The maximum stresses on a parabolic arch of a bridge are given by the roots of the equation 2,5 ·5r4 + 9r2 + 8r + 2 = 0, where r is the ratio of the length of the arch occupied by a moving load such as a train. Find r.

54. Graph of the Function f(ax). Before proceeding to the summary in the next section, we shall see how the graph of f(ax) may be found from that of f(x). This will complete the study we shall make of pairs of related functions and their graphs. Consider the

If we substitute 4 for x in the first function, and half of 4, namely 2, in the second, the results are both equal to 8. Hence the point (4,8) lies on the first graph and (2, 8) on the second, and the abscissa of the latter point is half that of the former.

If we substitute any value for x in the first function, and half that value in the second, the results will be the same, namely x2 - 6x. Hence if (x1, y1) is a point on the first graph the point (x1/2, y1) will be on the second.