CHAPTER XXXII.

INDETERMINATE COEFFICIENTS.

325. In Art. 148 it was proved that if a rational integral function of a vanish when x

α, it is divisible by х - a without remainder. Let the function be

1

a be performed.

and let the division by xIt is clear that the first term of the quotient, namely, the term of highest degree in x, will be ax-1; and since dividend = divisor x quotient,

1

.. f(x)=(xα) (αx21···).

Suppose that f(x) also vanishes when x=ß, [ẞ not =α], then the product (x −α) × (ax2-1+.....), and therefore either (x —α), or (ax2¬1 +.......), must vanish when x= a is not zero; hence (ax11+.....) vanishes when

But B

-1

B.

x= ß, and is therefore divisible by x B; and if the division be performed, the first term of the quotient will

be ax2-2.

-2

.. f(x) = (x — α) (x — ẞ) (ax2-2 + ...).

...

In general, if there be r values a, ß, y, d, of x for which f(x) vanishes, r repetitions of this process will obviously produce

ƒ (x) = (x − α) (x − ẞ) (x − y) (x — §) ..... (ax2-r + .....).

Finally, if ax + ban-1 + ... vanish for n different values

of x, it has n factors x

Ꮯ, Ꮖ ß, etc., and a final factor

axn−n, or a. Therefore, under this hypothesis,

f(x)= a(x-α) (x − ẞ) (x − y)......,

in which there are n binomial factors.

Two or more of these binomial factors may of course be identical. Should there be a factor of the 7th degree, say (x — α)", it is evident that the number of remaining factors will be n -r.

326. A rational integral function of the nth degree in x cannot vanish for more than n values of x, unless the coefficients of all the powers of x are zero.

For, if f(x), being of the form

vanish for the n values a, ß, y..., it must be equivalent to

If now we substitute for x any other value, k suppose, different from each of the n values a, ẞ, y, etc.; then since no one of the factors k — a, k — ß, etc., is

-2

zero,

their

and

continued product cannot be zero, and therefore ƒ (x) cannot vanish for the value x=k, except a itself is zero. But if a be zero, f(x) reduces to ban-1 +c2+ is of the (n-1)th degree; and hence it can only vanish for n 1 values of x, except b is zero. And so on. When all the coefficients are zero, the function will clearly vanish for any value whatever of x.

-

is equal to zero, are the roots of the equation

Hence, by Art. 326, an equation of the nth degree cannot have more than n roots, except the coefficients of all the different powers of the unknown quantity are zero, in which case any value of x satisfies the equation.

be equal to one another for more than n values of x, it follows that the equation

Hence, by Art. 327, the coefficients of all the different powers of x must be equal to zero.

Thus

a = p, b

= q, c=r, etc.

Hence, if two rational integral functions of the nth degree in x be equal to one another for more than n values of x, the coefficient of any power of x in one function is equal to the coefficient of the same power of x in the other.

This is the principle of indeterminate coefficients as applied to rational integral functions.

329. When any two rational integral expressions, which have a limited number of terms, are equal to one another for all values of the letters involved, the condition of the last article is clearly satisfied, for the number of values must be greater than the index of the highest power of any contained letter.

Hence when any two rational integral expressions, which have a limited number of terms, are equal to one another for all values of the letters involved in them, we may equate the coefficients of the different powers of any letter.

be equal to one another for all finite values of x for which they are convergent, then

ɑo = bo, ɑ1 = b1,•••, ɑ„=bn etc.

If there be a finite value of x for which the two series are convergent, they are also convergent for x=0 [Criterion IX., Art. 322], and when = 0, by the corollary of Art. 322,

x=

We now have, for the finite values of x that make the original series convergent,

But these two series are convergent for all the values of x that make a + ax + and b1+box+

...

...

conver

gent [Art. 322], and hence, by again putting ≈= 0, we