In algebra, the factor theorem connects polynomial factors with polynomial roots. Specifically, if is a polynomial, then is a factor of if and only if (that is, is a root of the polynomial). The theorem is a special case of the polynomial remainder theorem.[1][2]

The theorem results from basic properties of addition and multiplication. It follows that the theorem holds also when the coefficients and the element belong to any commutative ring, and not just a field.

In particular, since multivariate polynomials can be viewed as univariate in one of their variables, the following generalization holds : If and are multivariate polynomials and is independent of , then is a factor of if and only if is the zero polynomial.

Factorization of polynomials edit

Two problems where the factor theorem is commonly applied are those of factoring a polynomial and finding the roots of a polynomial equation; it is a direct consequence of the theorem that these problems are essentially equivalent.

The factor theorem is also used to remove known zeros from a polynomial while leaving all unknown zeros intact, thus producing a lower degree polynomial whose zeros may be easier to find. Abstractly, the method is as follows:[3]

  1. Deduce the candidate of zero   of the polynomial   from its leading coefficient   and constant term  . (See Rational Root Theorem.)
  2. Use the factor theorem to conclude that   is a factor of  .
  3. Compute the polynomial  , for example using polynomial long division or synthetic division.
  4. Conclude that any root   of   is a root of  . Since the polynomial degree of   is one less than that of  , it is "simpler" to find the remaining zeros by studying  .

Continuing the process until the polynomial   is factored completely, which all its factors is irreducible on   or  .

Example edit

Find the factors of  

Solution: Let   be the above polynomial

Constant term = 2
Coefficient of  

All possible factors of 2 are   and  . Substituting  , we get:

 

So,  , i.e,   is a factor of  . On dividing   by  , we get

Quotient =  

Hence,  

Out of these, the quadratic factor can be further factored using the quadratic formula, which gives as roots of the quadratic   Thus the three irreducible factors of the original polynomial are     and  

Proof edit

Several proofs of the theorem are presented here.

If   is a factor of   it is immediate that   So, only the converse will be proved in the following.

Proof 1 edit

This argument begins by verifying the theorem for  . That is, it aims to show that for any polynomial   for which   it is true that   for some polynomial  . To that end, write   explicitly as  . Now observe that  , so  . Thus,  . This case is now proven.

What remains is to prove the theorem for general   by reducing to the   case. To that end, observe that   is a polynomial with a root at  . By what has been shown above, it follows that   for some polynomial  . Finally,  .

Proof 2 edit

First, observe that whenever   and   belong to any commutative ring (the same one) then the identity   is true. This is shown by multiplying out the brackets.

Let   where   is any commutative ring. Write   for a sequence of coefficients  . Assume   for some  . Observe then that  . Observe that each summand has   as a factor by the factorisation of expressions of the form   that was discussed above. Thus, conclude that   is a factor of  .

Proof 3 edit

The theorem may be proved using Euclidean division of polynomials: Perform a Euclidean division of   by   to obtain   where  . Since  , it follows that   is constant. Finally, observe that  . So  .

The Euclidean division above is possible in every commutative ring since   is a monic polynomial, and, therefore, the polynomial long division algorithm does not involves any division of coefficients.

Corollary of other theorems edit

It is also a corollary of the polynomial remainder theorem, but conversely can be used to show it.

When the polynomials are multivariate but the coefficients form an algebraically closed field, the Nullstellensatz is a significant and deep generalisation.

References edit

  1. ^ Sullivan, Michael (1996), Algebra and Trigonometry, Prentice Hall, p. 381, ISBN 0-13-370149-2
  2. ^ Sehgal, V K; Gupta, Sonal, Longman ICSE Mathematics Class 10, Dorling Kindersley (India), p. 119, ISBN 978-81-317-2816-1.
  3. ^ Bansal, R. K., Comprehensive Mathematics IX, Laxmi Publications, p. 142, ISBN 81-7008-629-9.