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.
567. Number of Roots. Every equation of the nth degree has n roots, and no more.
Denote the given equation by f(x)= 0, where The equation f(x)= 0 has a root, real or imaginary ; let this be denoted by a_1; then f(x) is divisible by x - a_1, so that
where f_1(x) is a rational integral function of n -1 dimensions. Again, the equation f_1(x)=0 has a root, real or imaginary; let this be denoted by a_2; then f_1(x) is divisible by x-a_2, so that where f_2(x) is a rational integral function of n - 2 dimensions.
Thus .
Proceeding in this way, we obtain Hence the equation f(x)=0 has n roots, since f(x) vanishes when x has any of the values a_1, a_2, a_3, \ldots a_n.
Also the equation cannot have more than n roots; for if x has any value different from any of the quantities a_1, a_2, a_3, \ldots a_n, all the factors on the right are different from zero, and therefore f(x) cannot vanish for that value of x.
In the above investigation some of the quantities a_1, a_2, a_3, \ldots a_n, may be equal; in this case, however. we shall suppose that the equation has still n roots, although these are not all different.
568. Depression of Equations. If one root of an equation is known it is evident from the preceding paragraph that we may by division reduce or depress the equation to one of the next lower degree containing the remaining roots. So if k
