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52. The General Theorem. If
where is a positive integer, and any numbers, real or complex, independent of , to each value of corresponds a single value of .
We proceed to demonstrate that conversely to each value of corresponds a set of values of , i. e. that there are numbers which, substituted for in the polynomial , will give this polynomial any value, , which may be assigned.
It will be sufficient to prove that there are values of which render equal to 0, inasmuch as from this it would immediately follow that the polynomial takes any other value, , for values of ; viz., for the values which render the polynomial of the same degree, , equal to 0.
53. Root of an Equation. A value of for which is 0 is called a root of this polynomial, or more commonly a root of the algebraic equation
54. Theorem. Every algebraic equation has a root.
Given .
Let denote the modulus of . We shall assume, though this can be proved, that among the values of corresponding to all possible values of there is a least value, and that this least value corresponds to a finite value of .
Fig. 4.
Let denote this least value of , and the value of to which it corresponds. Then .
For if not, will be represented in the plane of complex numbers by some point distinct from the null-point .
Through draw a circle having its centre in the null-point . Then, by the hypothesis made, no value can be given which will bring the corresponding -point within this circle.
But the -point can be brought within this circle.
For, and being the values of and which correspond to , change by adding to a small increment , and let represent the consequent change in . Then is defined by the equation
On applying the binominal theorem and arranging the terms with reference to powers of , the right member of this equation becomes
Let be the complex number
expressed in terms of its modulus and angle, and the corresponding expression for . ThenThe point which represents for any particular value of can be made to describe a circle of radius about the null-point by causing to increase continuously from 0 to 4 right angles.
In the same circumstances the point representing
will describe an equal circle about the point and, therefore, come within the circle .But by taking small enough, may be made to differ as little as we please from ,24and, therefore, the curve traced out by (which represents , as runs through its cycle of values), to differ as little as we please from the circle of centre and radius .
Therefore by assigning proper values to and , the -point () may be brought within the circle .
The -point nearest the null-point must therefore be the null-point itself.25
55. Theorem. If be a root of , this polynomial is divisible by .
For divide by , continuing the division until disappears from the remainder, and call this remainder , the quotient , and, for convenience, the polynomial .
Then we have immediately
holding for all values of .Let take the value ; then vanishes, as also the product .
Therefore when , , and being independent of it is hence always 0.
56. The Fundamental Theorem. The number of the roots of the polynomial is .
For, by § 54, it has at least one root; call this ; then, by § 55, it is divisible by , the degree of the quotient being .
Therefore we have
Again, by § 54, the polynomial has a root; call this , and dividing as before, we have
Since the degree of the quotient is lowered by 1 by each repetition of this process, repetitions reduce it to the first degree, or we have
a product of factors, each of the first degree.
Now a product vanishes when one of its factors vanishes (§ 36, 3, Cor.), and the factor vanishes when , when when . Therefore vanishes for the values, , of .
Furthermore, a product cannot vanish unless one of its factors vanishes (§ 36, 4, Cor.), and not one of the factors , vanishes unless equals one of the numbers .
The polynomial has therefore and but roots.
The theorem that the number of roots of an algebraic equation is the same as its degree is called the fundamental theorem of algebra.
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