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the + placed before the leading element indicating the aggregate of all the elements which can be obtained from it by suitable interchanges of suffixes and adjustment of signs.
Sometimes the determinant is still more simply expressed by enclosing the leading element within brackets; thus (azb2c3d4...) is used as an abbreviation of E+a7b2c2d4 ...
Example. In the determinant (a1b2c3d4e5) what sign is to be prefixed to the element a4b3c1d5e2 ?
Here 3, 1, and 2 are inverted with respect to 4; 1 and 2 are inverted with respect to 3, and 2 is inverted with respect to 5; hence there are six inversions and the sign of the element is positive.
529. If in Art. 525, each of the constituents by, Cy ... k, is equal to zero the determinant reduces to a Aj; in other words it is equal to the product of a, and a determinant of the (n-1)th order, and we easily infer the following general theorem.
If each of the constituents of the first row or column of a determinant is zero except the first, and if this constituent is equal to m, the determinant is equal to m times that determinant of lower order which is obtained by omitting the first column and first row.
Also since by suitable interchange of rows and columns any constituent can be brought into the first place, it follows that if any row or column has all its constituents except one equal to zero, the determinant can immediately be expressed as a determinant of lower order.
This is sometimes useful in the reduction and simplification of determinants.
Example. Find the value of
11 20 38 6 3 0 9 11 -2 36 3
19 6 17 22 Diminish each constituent of the first column by twice the corresponding constituent in the second column, and each constituent of the fourth column by three times the corresponding constituent in the second column, and we obtain
8 11 20 5 0 3 0 0 15 -2 36 9
7 16 17 4 and since the second row has three zero constituents, this determinant
THEORY OF EQUATIONS.
GENERAL FORM OF AN EQUATION OF THE nth DEGREE.
530. LET Porn + P22n-1+porn-?+
+ Pn-1X + Pn be a rational integral function of x of n dimensions, and let us denote it by f(x); then f(x)=0 is the general type of a rational integral equation of the nth degree. Dividing throughout by Po, we see that without any loss of generality we may take
2n +P13n-1 +PqXn-2+. +Pn-1X+Pn=0 as the general form of a rational integral equation of any degree.
Unless otherwise stated the coefficients P1, P2, ... Pn will always be supposed rational. If any of the coefficients Pu P2,
Pn are zero, the equation is said to be incomplete, otherwise it is called complete.
531. Any value of x which makes f(x) vanish is called a root of the equation f(x)=0.
532. We shall assume that every equation of the form f(x)=0 has a root, real or imaginary. The proof of this proposition will be found in treatises on the Theory of Equations; it is beyond the range of the present work.
DIVISIBILITY OF EQUATIONS. 533. If a is a root of the equation f(x)=0, then is f(x) exactly divisible by x-a.
Divide the first member by x- a until the remainder no longer contains x. Denote the quotient by Q, and the remainder, if there be one, by R. Then we have
Now since a is a root of the equation x=a, therefore
Q(a-a) +R=0, hence
R=0; that is, the first member of the given equation is exactly divisible by x — a.
534. Conversely, if the first member of f(x)=0 is exactly divisible by x-a, then a is a root of the equation. For, the division being exact
Q(x-a)=0, and the substitution of a for x satisfies the equation; hence a is a root.
DIVISION BY DETACHED COEFFICIENTS.
535. The work of dividing one multinomial by another may be abridged by writing only the coefficients of the terms. The following is an illustration.
Example. Divide 3.25 – 8.4 – 5x3 +26x2 – 33x+26 by 203 — 222 -4x+8. 1+2+4-8)3–8– 5+26–33+26(3–2+3
5+ 2 Thus the quotient is 3x2 – 2x +3 and the remainder is – 5x+2.
It should be noticed that in writing down the divisor, the sign of every term except the first has been changed ; this enables us to replace the process of subtraction by that of addition at each successive stage of the work.
HORNER'S METHOD OF SYNTHETIC DIVISION.
536. The work of division by detached coefficients may be abridged by the following arrangement, which is known as Horner's Method of Synthetic Division.
Let us consider the example of the preceding article. The arrangement of the work is as follows:
1 3-8- 5+26 – 33 +26
3–2+ 3+ 0 – 5+ 2 [Explanation. The column of figures to the left of the vertical line consists of the coefficients of the divisor, the sign of each after the first being changed ; the second horizontal line is obtained by multiplying 2, 4, -8 by 3, the first term of the quotient. We then add the terms in the second column to the right of the vertical line; this gives — 2, which is the coefficient of the second term of the quotient. With the coefficient thus obtained we form the next horizontal line, and add the terms in the third column; this gives 3, which is the coefficient of the third term of the quotient.
By adding up the other columns we get the coefficients of the terms in the remainder.]
537. In employing this method in the following articles our divisor will be of the form xa, which enables us to still further simplify the work as the following example shows :
Example. Find the quotient and remainder when 3x7 — 206 +3124 +213+5 is divided by +2.
3 - 1 0 31 0 0 21 5 1-2
-6 14 - 28 -6 12 - 24
Thus the quotient is 3x6 — 725 +14x4 + 3x3 — 6x2 + 12x-3, and the remainder is 11.
[Explanation. The first horizontal line contains the coefficients of the dividend, zero coefficients being used to represent terms corresponding to powers of x which are absent. The divisor is written at the right of this line with its sign changed (Art. 535) and 1, the coefficient of x omitted. The first term of the third horizontal line, which contains the quotient, is the result of dividing 3, the coefficient of x7 in the dividend, by 1, the coefficient of x in the divisor. This is then multiplied by the divisor – 2 and the result is – 6, the first term of the second horizontal line; the sum of -1 and -6 gives — 7, the second term of the quotient, which multiplied by -2 gives 14 for the second term of the second horizontal line; the addition of 14 and 0 gives 14 for the third term of the quotient, which multiplied by–2 gives –28 for the third term of the second line, and so on.]