Factor theorem

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]

Factorization of polynomialsEdit

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. "Guess" a zero   of the polynomial  . (In general, this can be very hard, but maths textbook problems that involve solving a polynomial equation are often designed so that some roots are easy to discover.)
  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  .


Find the factors of


To do this one would use trial and error (or the rational root theorem) to find the first x value that causes the expression to equal zero. To find out if   is a factor, substitute   into the polynomial above:


As this is equal to 18 and not 0. This means   is not a factor of  . So, we next try   (substituting   into the polynomial):


This is equal to  . Therefore  , which is to say  , is a factor, and   is a root of  

The next two roots can be found by algebraically dividing   by   to get a quadratic:


and therefore   and   are factors of   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.