CHAPTER XLII.
UNDETERMINED COEFFICIENTS.
FUNCTIONS OF FINITE DIMENSIONS.
481. In Art. 105 it was proved that if any rational integral function of 2 vanishes when x=a, it is divisible by x-a.
Let
be a rational integral function of x of x dimensions, which vanishes when x is equal to each of the unequal quantities
Denote the function by f(x); then since f(x) is divisible
by x-a_1, we have
the quotient being of n-1 dimensions.
Similarly, since f(x) is divisible by x-a_2, we have
the quotient being of n-2 dimensions; and
Proceeding in this way, we shall finally obtain after n
divisions
482. If a rational integral function of m dimensions vanishes
for more than n values of the variable, the coefficient of each
power of the variable must be zero.
Let the function be denoted by f(x), where
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