How We Can Reduce The Number Of Dimensions Of An Equation

How one can reduce the number of dimensions of an equation when one knows one of its roots and How one can examine whether a given quantity is the value of a root

The sum of an equation which contains several roots can always be divided by a binomial composed of the unknown quantity minus the value of one of the true roots—whichever it may be—or plus the value of one of the false roots; by this means, its degree is reduced accordingly.

And conversely, if the sum of an equation cannot be divided by a binomial composed of the unknown quantity plus or minus some other quantity, this shows that the other quantity is not the value of any of its roots.

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The above can be divided by x−2x−2, and by x−3x−3, and by x−4x−4, and by x+5x+5, but not by x+x+ or −− any other quantity, this shows that it can only have the four roots 2, 3, 4, and -5.

One also sees from this how many true roots and how many false ones there can be in each equation: namely, there can be as many true roots as there are sign changes between + and −, and as many false ones as there are instances where two + signs or two − signs follow one another.

As in the last case, because after +x4+x4 there is −4x3−4x3 (a change from + to −), and after −19x2−19x2 there is +106x+106x, and after +106x+106x there is −120−120, which are two more changes, one knows that there are three true roots; and one false root, because of the two − signs of −4x3−4x3 and −19x2−19x2 that follow one another.

Moreover, it is easy to make all the roots that were false become true, and likewise make all the ones that were true become false, by changing all the + or − signs that are in the second, fourth, sixth, or other places indicated by even numbers, without changing those in the first, third, fifth, and similar positions indicated by odd numbers.

Instead of +x4−4x3−19x2+106x−120=0,

we write

+x4+4x3−19x2−106x−120=0,

we have an equation in which there is only one true root, which is 5, and three false roots, which are 2, 3, and 4.

If, without knowing the value of the roots of an equation, we want to increase or decrease any given quantity, we need only suppose, in place of the unknown, another quantity that is more or less than it, and substitute it everywhere.

As for example, if we want to increase the root of this equation by 3:

x4+4x3−19x2−106x−120=0,

put y in the place of x, let y exceed x by 3, so that y−3=x.

Then instead of x2, we write the square of y−3 which is y2−6y+9

Instead of x3, we write its cube: y3−9y2+27y−27

Instead of x4, we write the fourth power:

y4−12y3+54y2−108y+81

Thus, substituting y everywhere instead of x, we obtain:

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or

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whose true root is now 8 instead of 5, since it has been increased by 3.

If, on the other hand, it is desired to diminish by 3 the roots of the same equation, we must put y±3 = x and y²−6y+9 = x², and so on.

So that instead of x⁴ + 4x³ − 19x² − 106x − 120 = 0, we have

y⁴ + 12y³ + 54y² + 108y + 81 + 4y³ + 36y² + 108y + 108 − 19y² − 114y − 171 − 106y − 318 − 120 ————————————————— y⁴ + 16y³ + 71y² − 4y − 420 = 0.

Increasing the true roots of an equation diminishes the false roots by the same amount; and on the contrary diminishing the true roots increases the false roots; while diminishing either a true or a false root by a quantity equal to it makes the root zero; and diminishing it by a quantity greater than the root renders a true root false or a false root true. Thus by increasing the true root 5 by 3, we diminish each of the false roots, so that the root pre- viously 4 is now only 1, the root previously 3 is zero, and the root previously 2 is now a true root, equal to 1, since −2+3 = +1. This explains why the equation y²−8y²−y+8 = 0 has only three roots,

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