Factor theorem

From Wikipedia, the free encyclopedia

In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.[1]

The factor theorem states that a polynomial has a factor if and only if (i.e. is a root).[2]

More generally, a bivariate polynomial has a factor if and only is the zero polynomial. The above theorem is the case where

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 its all factors is irreducible on or .


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 =


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


  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.