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 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 at

To do this you 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, which can be solved directly, by the factor theorem or by the quadratic formula.

and therefore and are the factors of


  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 .