(107.) Now we may reverse the above process, that is, we may extract the cube root of a polynomial by the following

'RULE.

I. Having arranged the terms of the polynomial according to the powers of some one of the letters, seek the cube root of the first term, which place at the right of the polynomial for the first term of the root, also place it at the left by itself, for the first term of a column, headed FIRST COLUMN. Then multiply it into itself, and place the product for the first term of a column, headed SECOND COLUMN. Again, multiply this last result by the same first term of the root and subtract the product from the first term of the polynomial, and then bring down the next three terms of the polynomial, for the

Add the first term of the root just found to the first term of the first column, the sum will constitute its second term, which must be multiplied by the first term of the root, and the result added to the first term of the second column, for its second term, which we will call the FIRST TRIAL DIVISOR. first term of the root must be added to the second term of the first column, forming its third term.

FIRST DIVIDEND.

The same

II. Divide the first term of the first dividend by the first term of the trial divisor, the quotient must be added to the root already found, for its second term, it must also be added to the last term of the first column, the result will be its fourth term, which must be multiplied by the second term of the root and the product added to the last term of the second column, which sum will give its third term, which in turn must be multiplied by the second term of the root, and the product subtracted from the first dividend.

III. To the remainder bring down three or four of the next terms of the polynomial for a SECOND DIVIDEND. Proceed with this second term of the root, precisely as was done with the first term, and so continue until the entire polynomial has been exhausted.

EXAMPLES.

1. What is the cube root of a* +3aPb+3ab2 +63 +3(a+b)*c+3(a+b)c? +c?

OPERATION.

ROOT.

3

FIRST COLUMN.

SECOND COLUMN.

3

a

a

a’ +3a+b+3ab2 +63 +3(a+b)*c+3(a+b)c<+co(a+b+c
a 2
2a
3a2

3aPb+3ab2 +63
За
3a2+3ab+62

3a2b+3ab2 +63 3a+b

3a? +6ab+352 Зa+2b or 3(a+b)2

3(a+b)*c+3(a+b)c2 +c8 $ 3a+36 3(a+b)2 +3(a+b)c+c?

3(a+b)*c+3(a+b)c2+c8
2 or 3(a+b)

0
3(a+b)+c
2. What is the cube root of 8x +36x2 +54x+27 ?

2

ROOT.

9x2

27x-18x3+ 4x2

6x5

-40x3

4

5. What is the cube root of the polonomial
x6-6x+15x1-20x3+15x2-6x+1?

Ans. x2-2x+1.

6. What is the cube root of the polynomial
a9+12a5x2-8a3x3-6a7x?

Ans. a3-2ax.

7. What is the cube root of

a6-3a5+6a4-7a3+6a2-3a+1?

8. What is the cube root of

5

3

Ans. a2-a+1.

x+6x+21x+44x3 +63x2+54x+27?

6

Ans. x2+2x+3.

(108.) From the above rule, for extracting the cube root of a polynomial, we can easily deduce the rule which we have given in the Higher Arithmetic for the extraction of the cube root of a number.

This rule is also particularly interesting because of its close analogy to the method of finding the numerical roots of a cubic equation, as explained in a subsequent part of this work.

IRRATIONAL OR SURD QUANTITIES.

(109.) AN IRRATIONAL QUANTITY, OR SURD, is a quantity affected with a fractional exponent or radical, without which, it can not be accurately expressed.

Thus,

√3 is a surd, since the square root of 3 can not be accu

rately found; also 82, 43, 4, 5, &c., are surd quanti