NATURE AND SOLUTION OF EQUATIONS OF ALL DEGREES.

I. Formation of Equations and Properties of their Roots. 125. Every equation containing only one unknown quantity is either of the form

+ tx + u = 0.

(1),

x2+p x2-1+q x2-2 + where n is a whole positive number, or if it be not of this form, it can easily be reduced to it by fundamental operations. Let X denote the first member of eq. (1); and if X or x2+p x2 + etc., be divided by a, the quotient will be

x

x2-1 + (a + p) x2-2 + (a2 + pa + q) x2-3+ etc. and the remainder

+ta + u

(3),

a"+pa”-1 + qa”-2 + will be precisely of the form of the original polynomial X, the only change being that a is substituted for x. The result thus far is entirely independent of the value of a, but if a be a root of the equation (1) then by Art. 83, the remainder (3) is equal to zero, and the original equation (1) is divisible by x a without remainder, the quotient (2) being a polynomial of the degree n 1. We have therefore

-

-2

X = (x − a) {x3-1 + (a +p) x"−2 + etc.} = 0 . . . . (4), and this equation is evidently satisfied if either

-2

x — a = 0 or x2¬1 + (a + p) x2-2 + etc. Again, let the equation (a + p) x2-2 + etc.

-

-2

=

= € 0.

O be divided by b; then if b is a root of this equation, it may be shown as before that the remainder will be zero, and the quotient will be of the degree n2; consequently

x+(a+p) x-2+ etc. = (x − b) (x2 + etc.).

Substituting this in equation (4) we get

X = (x − a) (x — b) (x2 + etc.) = 0. The proposed equation will consequently be satisfied if

Proceeding in this manner till the powers of x are exhausted, it will be scen that the original equation is divisible successively by x-a, x — b, . . . xl, where a, b, c, . . . . l are the roots of the equation ; and as the last of these divisors, and the last of the equations will involve only the single powers of x, the equation will admit of no further division. Hence

X = (x − a) (x − b) (x − e) (xd).... (x − 1), and the first member of equation (1) is resolved into n factors of the first degree. The original equation X 0 will be satisfied, if any of the factors x -a, xb, etc., be equal to zero, and in no other case. Hence each of the n quantities a, b, c, 1, when substituted for x in equation (1) will verify that equation, and it therefore follows that an equation has necessarily as many roots as there are units in the index of the highest power of the unknown quantity, and that it can have no greater number.

Conversely, if the roots of an equation be given, the equation will be

b, x

c, x

formed by the multiplication of certain factors obtained by connecting the roots severally, with their signs changed, to the unknown quantity. Thus if x d, etc., be the factors, then taking only one of these as x a, and putting it equal to 0, we get the simple equation

a, x

Take the product of the two factors xa, x — b; and we get in like manner the quadratic equation

x2 - (a + b) x + a b = 0.

(2). The product of three factors xa, x — b, x — e, will give the cubic

equation

-

x2 - (a + b + c + d) x3 + (ab + ac + ad+be+bd+cd) x2

- (abc + abd + acd+bcd)x + a b c d = 0 . . (4),

and so on to any extent.

The quantities a, b, c, d are called roots of the equation, a simple equation having only one root, a quadratic equation two roots, a cubic equation three roots, a biquadratic equation four roots, and generally an equation of the nth degree has n roots.

If one of the quantities a, b, c, d, etc. be of the form h+k; then another of them must be of the form h k; because the product of the two factors x (k + k) and · (h — √√ k), viz., 2 h x +hk does not involve either a surd or an imaginary quantity; hence surd and imaginary roots must occur in pairs. If k is positive, the two roots h + √k and hk will be real, and if k is negative, they will be imaginary.

From this method of formation of an equation we may deduce the following useful relations between the roots of the equation and the coefficients of the several terms.

(1). The coefficient of the second term is equal to the sum of the roots with their signs changed.

(2). The coefficient of the third term is equal to the sum of the products of the roots, taken two together, with their signs changed.

(3). The coefficient of the fourth term is equal to the sum of the products of the roots, taken three together, with their signs changed, and so on.

(4). The last term is equal to the product of all the roots with their signs changed.

Hence we see that

(a). If the second term of an equation is wanting, the sum of the positive roots is equal to the sum of the negative roots.

(B). If the signs of the terms of an equation be all positive, the roots will be negative; and if the signs be alternately positive and negative, the roots will be positive.

(7). Every root of an equation is a divisor of the last term.

EXAMPLES.

1. Form the cubic equation whose roots are 1, 2, and 5.

-

Ans. x+2x2. 13x+10=0.

126. The transformation of an equation is the changing it into another of the same degree whose roots shall have a specified relation to the roots of the proposed equation. Thus, if the general equation of the n' degree be

then if

. (1);

y be substituted for x in this equation it will be transformed into another whose roots are the same as those in (1), but with contrary signs, for y=

1

x.

If be substituted for x, then the roots of the new equation in y will

y

1

be the reciprocals of those of eq. (1), for y =—.

If be substituted for x, the roots of the new equation will be m

m

times those of the equation (1), for y = mx.

Ify be substituted for x, the roots of the new equation, freed from radicals, will be the squares of those of eq. (1), for y = x2.

But the most important transformation of all is the substitution of yr for x; for since y = x − r, the roots of the equation in y will be less than those of the given equation in x by the quantity r. Thus, if it be required to transform the equation x + px3 + q x2 + s x + t = 0 into another whose roots shall be less than those of the given equation by r, we have only to substitute y+r for x; hence we have

(y+r)*+ p (y + r)3 + q (y + r)2 + s (y + r) + t = 0, or y' + 4r y3 +62 y2+4y+r+pr3+qr2+sr+ t = 0.....(2), + P + 3p 7 + 3 pr2

+ 9

+2qr + S

which is the transformed equation required. But this operation may be much simplified in practice; for if x+px3 + q x2 + s x + t be actually divided by xr, the quotient will be

x2 + (r+p) x2 + (p2 + pr+ q) x + r22 + pr2 + qr + 8, and the remainder will be identically the same as the last term of (2). If this new quotient be divided by x- r, the quotient will be

x2 + (2 r + p) x + (3 r2 + 2 p r +q),

and the remainder will be the coefficient of y in equation (2). The last quotient divided by x- -r will give x + (3r+p) for quotient, and the remainder will be the coefficient of y2 in (2). Lastly, divide x + (3r+p) by x - r, and the remainder will be 4r+p, the coefficient of y3 in (2). This leads us to an elegant, simple, and systematic process for effecting the solution of numerical equations of all degrees.

If yr be substituted for x in (1), the roots of the new equation in y will be greater than those of (1) by r.

1. Transform the equation

EXAMPLES.

8x+12x2 - 5 x + 4 = 0 into another whose roots shall be less by 2 than the roots of the given equation.

*

Here we must substitute y +2 for x, or divide successively by x 2 by the synthetic process." Taking the latter method, we have this operation, viz.:

1 -8 +12
2-12

-6 0
2

-

8

5 + 4 (2
10

0

5 6

16

8 -21

4

-12

Hence y 12 y21 y 60 is the transformed equation, and its roots are each less by 2 than those of the given equation. It will be noticed that the transformed equation does not contain the third power of y, or that the second term of the equation has been removed. This is obvious, since the sum of the roots of the given equation is 8, the coefficient of the second term with its sign changed, and that each root has been diminished by 2, therefore the four roots have been diminished by 8, and the sum of the positive and negative roots of the transformed equation must be equal to 0, and the second term absent.

2. Diminish the roots of the equation x3- 6 x2 + 18 x

* The synthetic mode of division is simply a change of the arrangement of the common operation, and the conversion of the process of subtraction into that of addition, by changing the signs of all the terms of the divisor except that of the first term. Thus if it were required to divide a 8a3+12 a 5x + 4 by ≈ - 2, the coefficients of the several terms are written down with their proper signs, and the second term of the divisor (x-2) is written with its sign changed on the right of the coefficients. Then the coefficient of the first term of the divisor being unity, the coefficient of the first term of the dividend is the coefficient of the first term of the quotient; multiplying 1 by 2, the product 2 is written below 8, and added thereto to give 6, the coefficient of the second term of the quotient; then 6 is multiplied by 2 which gives 12, and this product is written below +12, and added thereto, giving 0 for the coefficient of the third term, and so on, as in the following process:

will be readily comprehended by comparing the work in this example with the usual mode of division.

3. Find the equation whose roots shall be less by 4 than those of the equation x3 15x+81x- 243 = 0.

Ans. y

3y+9y950.

4. Find the equation whose roots shall be greater by 1 than those of the equation * - 4x3- 8 x + 32 = 0.

Ans. y 8 y3 + 18 y 24 y +45=0. 5. Find the equation whose roots shall be less by 3 than those of the equation +9 x3 + 24 x2 + 18 x Ans. y + 10 2 y3 + 32·64 y2 + 34·938 y — 3.1889 = 0.

110.

III. Solution of Equations.

127. We have seen (125, y) that every root of an equation is a factor of the last term, and therefore, in the case of integral roots, these may often be easily found by trial. Thus, in the equation

x3 + 3x2+9x — 38 = 0

the integral factors of the last term are ± 1, ± 2, 19, 38. It is easily seen that neither + 1 nor will verify the equation; but - 1 trying2, it is found to succeed;

But this is best effected by dividing the first side of the equation by x2, in the following manner, where the synthetic mode is adopted:

The remainder being 0, shows that 2 is a root of the equation, and also that the numbers 1, 5, 19 are the coefficients of the depressed equation containing the remaining roots.

For since 2 is a root of the equation, it is evident that x 2 is one of the factors of the expression 2+3x2 + 9 x 38, and performing the division of this last expression by x 2, either by the common process or the synthetic process above, we get

(x2) (x2+5x+19)=x3 + 3 x2 + 9 x − 38;

... (x – 2) (x +5 + 19)=0.

To satisfy this equation we must have either x 2 = 0, or 2+5x+19 = 0; the former gives x = 2, the root already found, and the quadratic x+5x+19 0, will give the two other roots, which in this case are found to be imaginary. Hence if one root of an equation be known, the equation may be depressed to one of the next lower degree.

1. If x3

EXAMPLES.

12+4x+2070, what are the values of x?