The rule given in this chapter is inserted on account of its excellence as an exercise in computation. The examples chosen will require but little use of algebraical signs, that they may be understood by those who know no more of algebra than is contained in the present work.
To solve an equation such as
2x⁴ + x² - 3x = 416793,
or, as it is usually written,
2x⁴ + x² - 3x - 416793 = 0,
we must first ascertain by trial not only the first figure of the root, but also the denomination of it: if it be a 2, for instance, we must know whether it be 2, or 20, or 200, &c., or ·2, or ·02, or ·002, &c.
This must be found by trial; and the shortest way of making the trial is as follows: Write the expression in its complete form. In the preceding case the form is not complete, and the complete form is
2x⁴ + 0x³ + 1x² - 3x - 416793.
To find what this is when x is any number, for instance, 3000, the best way is to take the first multiplier (2), multiply it by 3000, and take in the next multiplier (0), multiply the result by 3000, and take in the next multiplier (1), and so on to the end, as follows:
2 × 3000 + 0 = 6000; 6000 × 3000 + 1 = 18000001
18000001 × 3000 - 3 = 54000002997
54000002997 × 3000 - 416793 = 162000008574207
Now try the value of the above when x = 30. We have then, for the steps, 60 (2 × 30 + 0), 1801, 54027, and lastly,
1620810 - 416793,
or x = 30 makes the first terms greater than 416793. Now try x = 20 which gives 40, 801, 16017, and lastly,
320340 - 416793,
or x = 20 makes the first terms less than 416793. Between 20 and 30, then, must be a value of x which makes 2x⁴ + x²-3x equal to 416793. And this is the preliminary step of the process.
Having got thus far, write down the coefficients +2, 0, +1,-3, and -416793, each with its proper algebraical sign, except the last, in which let the sign be changed. This is the most convenient way when the last sign is-.