Factor theorem

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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 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. "Guess" a zero of the polynomial . (In general, this can be very hard, but math 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.