and suppose that f(x) vanishes when a is equal to each of the unequal values a_1, a_2, a_3, a_n; then
Let c be another value of 2 which makes f(x) vanish; then
since f(c)=0, we have
and therefore p_0= 0, since, by hypothesis, none of the other
factors is equal to zero. Hence f(x) reduces to
By hypothesis this expression vanishes for more than n
values of x, and therefore p_1 = 0.
In a similar manner we may show that each of the coefficients p_2, p_3, p_n, must be equal to zero.
This result may also be enunciated as follows:
If a rational integral function of n dimensions vanishes for more than n values of the variable, it must vanish for every value of the variable.
Cor. If the function f(x) vanishes for more than n values of x, the equation f(x)= 0 has more than n roots.
Hence also, if an equation of n dimensions has more than n roots it is an identity.
Ex. Prove that
(x-b)(x-c) (a-b)(a-c) + (x-c)(x-a) {(b-c)(b-a)} + (x-a)(x-b) {(c-a)(c-b)} = 1
This equation is of two dimensions, and it is evidently satisfied by each of the three values a, b, c; hence it is an identity.
483. If two rational integral functions of n dimensions are equal for more than n values of the variable, they are equal for every value of the variable.
Suppose that the two functions
are equal for more than n values of x; then the expression
