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When a power of x is wanting in the proposed equation, we must supply its place with a cipher.

Ex. 3. Find all the roots of the cubic equation

x3-7x=-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.

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4.0696 -1.47728128=5th div. 141586432=5th div'd.

4.07049 -1.4769149359

132922344231

4.07058-1.4765485837-6th div. 8664087769-6th div'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 x3-7x+7=0

to a quadratic equation by dividing by x-1.356895867, and we shall obtain

x2+1.356895867x-5.158833606=0.

Solving this equation, we obtain

=-.678447933±√5.619125204.

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Ex. 4. Find a root of the equation 2x3+3x2-850.

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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

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38.410

38.4165 98.99233995

98.9040 3d divisor. 137920- 3d dividend.

98.980815= 4th divisor. 38977595= 4th div'd.

38.4180 99.00386535=5th div'r. 9279893015=5th div'd.

98942405

29697701985

Hence one root of the equation

is -2.16139.

5x3-6x2+3x=-85

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

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6.240246 12.980235860667=4th d. 10683412068213=4th d.

Ex. 7. Find all the roots of the equation

x3-15x2+63x-50=0.

1088=2d dividend.
1038375936512

49624063488=3d div. 38940651419787

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Ans. 364.

Ex. 9. Extract the cube root of 48228544.

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?

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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

x2-8x3+14x2+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:

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12.6 51.44 66.193068=3d div. .40359759-3d div'd.

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and by division we obtain the four figures 0679.

The other three roots may be found in the same manner.

Hence the four roots are

Ex. 2. Find a root of the equation

x2+2x2+3x3+4x2+5x=20.

.7320508,

.7639320,

2.7320508,

5.2360679.

We have found, by Sturm's Theorem, that this equation has

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Dividing the fourth dividend by the fourth divisor, we ob

tain the figures 789.

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