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.
563. 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.
564. The work of dividing one multinomial by another may be abridged by writing only the coefficients of the terms. The following is an illustration.
Ex. Divide 3x^5-8x^4-5x^3+ 26x^2-33x+26 by x^3-2x^2-4x+8.
1+2+4+4-8)3-8-5+26-33 +26(3-2+3 3+ 6 + 12 - 24
-2+ 7+ 2-33 -2- 4- 8+16
3- 6-17+ 26 3+ 6+12-24
-5+ 2
Thus the quotient is 3x^2 - 2x +3 and the remainder is -5x + 2.
It should be noticed that in writing 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.
565. For convenience we again give an explanation of Horner’s Method of Synthetic Division, which has already been considered in Art. 63.
