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When a power of w is wanting in the proposed equation, we must supply its place with a cipher. Ex. 3. Find all the roots of the cubic equation

23-7x=-7. The work of the following example is exhibited in an abbreviated form. Thus, when we multiply A by r, and add the product to B, we set down simply this result. We do the same in the next column, thus dispensing with half the number of lines employed in the preceding example. Moreover, we may omit the ciphers on the left of the successive dividends, if we pay proper attention to the local value of the figures. Thus it will be seen that in the operation for finding each successive figure of the root, the decimals - under B increase one place, those under C increase two places, and those under V increase three places. 1+0 -7

-7 (1.356895867 =X. 1 -6

- 6 2 -4=1st div'r. _1=1st dividend. 3.3 -3.01

-.903 3.6 -1.93=2d diy'r. -97=2d dividend. 3.95 -1.7325

86625 4.00 -1.5325=3d div'r. 10375= 3d dividend. 4.056 -1.508161

9048984 4.062 -1.483792=4th div'r. 1326016= 4th dividend. 4.0688 -1.48053696

1184429568 4.0696 -1.47728128=5th div. 141586432=5th div'd. 4.07049 -1.4769149359

132922344231 4.07058 - 1.4765485837=6th div. 8664087769=6th diy'd. Having proceeded thus far, four more figures of the root, 5867, are found by dividing the sixth dividend by the sixth divisor.

We may find the two remaining roots by the same process; or, after having obtained one root, we may depress the equation

03-7+7=0 to a quadratic equation by dividing by 2-1.356895867, and we shall obtain

22 +1.356895867x-5.158833606=0.

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Solving this equation, we obtain
x=-.678447933+ V5.619125204.

-3.048917, Hence the three roots are

1.356896,

1.692021. Ex. 4. Find a root of the equation 2x3 + 3x2=850. 2+3 +0

=850 (7.0502562208 17 119

833 31 336=1st divisor. 17=1st dividend. 45.10 338.2550

16.912750 45.20 340.5150=2d divisor. 87250=2d dividend. 45.3004 340.52406008

68104812016 45.3008 340.53312024= 3d div. 19145187984= 3d div'd. 45.30130 340.5353853050

17026769265250 45.30140 340.5376503750=4th d. 2118418718750=4th div.

Dividing the fourth dividend by the fourth divisor, we obtain the figures 62208, which make the root correct to the tenth decimal place.

The two remaining values of x may be easily shown to be imaginary.

When a negative root is to be found, we change the signs of the alternate terms of the equation, Art. 442, and proceed as for a positive root.

Ex. 5. Find a root of the equation 5x3 — 6x2 + 3x=-85.
Changing the signs of the alternate terms, it becomes

5x + 6x2 + 3x= +85. 5 +6

+85 (2.16139. 16 35

70 26 87=1st divisor. 15=1st dividend. 36.5 90.65

9.065 37.0 94.35=2d divisor. 5.935=2d dividend. 37.80 96.6180

5.797080 38.10 98.9040=3d divisor. 137920= 3d dividend. 38.405 98.942405

98942405 38.410 98.980815= 4th divisor. 38977595=4th div'd. 38.4165 98.99233995

29697701985 38.4180 99.00386535= 5th diy'r. 9279893015=5th diy'd.

+3

Hence one root of the equation

5x3-6x2 + 3x= -85 is -- 2.16139.

The same method is applicable to the extraction of the cube root of numbers.

Ex. 6. Let it be required to extract the cube root of 9; in other words, it is required to find a root of the equation

=9.

+0

1+0

=9 (2.0800838.
2
4

8 4

12=1st divisor. 1=1st dividend.
6.08 12.4864

.998912
6.16 12.9792=2 divisor. 1088=2d dividend.
6.24008 12.9796992064

1038375936512 6.24016 12.9801984192=3d d. 49624063488=3d div. 6.240243 12.980217139929 38940651419787 6.240246 12.980235860667=4th d. 10683412068213=4th d. Ex. 7. Find all the roots of the equation 23 - 15x2+633-50=0.

1.02804. Ans. 6.57653.

7.39543. Ex. 8. Find all the roots of the equation 23 +9.x2 +24x+17=0.

-1.12061. Ans. -3.34730.

-4.53209. Ex. 9. Extract the cube root of 48228544.

Ans. 364. Ex. 10. There are two numbers whose difference is 2, and whose product, multiplied by their sum, makes 100. What are those numbers ?

Ex. 11. Find two numbers whose difference is 6, and such that their sum, multiplied by the difference of their cubes, may produce 5000.

Ex. 12. There are two numbers whose difference is 4; and

the product of this difference, by the sum of their cubes, is 3400. What are the numbers ?

Ex. 13. Several persons form a partnership, and establish a certain capital, to which each contributes ten times as many dollars as there are persons in company. They gain 6 plus the number of partners per cent., and the whole profit is $392. How many partners were there?

Ex. 14. There is a number consisting of three digits such that the sum of the first and second is 9; the sum of the first and third is 12; and if the product of the three digits be increased by 38 times the first digit, the sum will be 336. Required the number.

636, Ans. or 725,

or 814. Ex. 15. A company of merchants have a common stock of $4775, and each contributes to it twenty-five times as many dollars as there are partners, with which they gain as much per cent. as there are partners. Now, on dividing the profit, it is found, after each has received six times as many dollars as there are persons in the company, that there still remains $126. Required the number of merchants.

Ans. 7, 8, or 9.

EQUATIONS OF THE FOURTH AND HIGHER DEGREES. 468. It may be easily shown that the method here employed for cubic equations is applicable to equations of every degree. For the fourth degree we shall have one more column of products, but the operations are all conducted in the same manner, as will be seen from the following example. Ex. 1. Find the four roots of the equation

24 -8.3 +14.x2 +4x=8. By Sturm's Theorem, we have found that these roots are all real; three positive, and one negative.

We then proceed as follows:

1-8
+14 + 4

=8 (5.2360679. -3 1

1

-5 +2 + 9 +44=1st divisor. 13=1st dividend. 7 44 53.288

10.6576 12.2 46.44 63.072=2d div'r. 2.3424=2d dividend. 12.4 48.92 64.626747

1.93880241. 12.6 51.44 66.193068=3d div. .40359759=3d div'd. 12.83 51.8249 66.509117736 .399054706416 12.86 52.2107 66.825633024=4th d. 4542883584=4th d. 12.89 52.5974 12.926 52.674956

12.932 52.752548 and by division we obtain the four figures 0679. The other three roots may be found in the same manner.

'- .7320508, Hence the four roots are

.7639320, 2.7320508,

5.2360679. Ex. 2. Find a root of the equation

203 +224 + 3x3+4x2+5x=20. We have found, by Sturm's Theorem, that this equation has a real root between 1 and 2.

We then proceed as follows: 1+2 +3 +4 +5

+ 20 (1.125789. 3 6

10
15

15 4 10

20

35=1st divisor. 5=1st dividend. 5 15

35
38.7171

3.87171 6 21

37.171 42.6585=2d divisor, 1.12829=2d dividend. 7.1 21.71 39.414 43.5027 2016

.87005 44032 7.2 22.43 41.730 44.3566 2080, 3d div'r. .25823|55968, 3d div'd. 7.8

23.16 42.211 008 44.5731 44750625 .22286|5723753125 7.4 23.90 42.695 032 44.7902 83203125, 4th d. 3536 9873046875, 4th do 7.5 2 24.05/04 43.182 080 7.54 24.2012 43.304790125 7.5 6 24.35 24 43.427 690500 7.58 24.50 40 7.605 24.54 2025 7.6 10 24.58|0075

Dividing the fourth dividend by the fourth divisor, we obtain the figures 789.

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