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 f(x) has a factor (x - k) if and only if f(k)=0 (i.e. k is a root).[2]
Factorization of polynomials
Main article: Factorization of polynomials

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]

"Guess" a zero a of the polynomial f. (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.)
Use the factor theorem to conclude that (x-a) is a factor of f(x).
Compute the polynomial g(x) = f(x) \big/ (x-a) , for example using polynomial long division or synthetic division.
Conclude that any root x \neq a of f(x)=0 is a root of g(x)=0. Since the polynomial degree of g is one less than that of f, it is "simpler" to find the remaining zeros by studying g.

Example

Find the factors at

\( x^3 + 7x^2 + 8x + 2. \)

To do this you would use trial and error to find the first x value that causes the expression to equal zero. To find out if (x - 1) is a factor, substitute x = 1 into the polynomial above:

\( x^3 + 7x^2 + 8x + 2 = (1)^3 + 7(1)^2 + 8(1) + 2 \)
= 1 + 7 + 8 + 2
= 18.

As this is equal to 18 and not 0 this means (x - 1) is not a factor of \( x^3 + 7x^2 + 8x + 2 \). So, we next try (x + 1) (substituting x = -1 into the polynomial):

\( (-1)^3 + 7(-1)^2 + 8(-1) + 2. \)

This is equal to 0. Therefore x-(-1), which is to say x+1, is a factor, and -1 is a root of x^3 + 7x^2 + 8x + 2.

The next two roots can be found by algebraically dividing \) x^3 + 7x^2 + 8x + 2 \) by (x+1) to get a quadratic, which can be solved directly, by the factor theorem or by the quadratic formula.

\( {x^3 + 7x^2 + 8x + 2 \over x + 1} = x^2 + 6x + 2 \)

and therefore (x+1) and \( x^2 + 6x + 2 \) are the factors of \( x^3 + 7x^2 + 8x + 2.\(
References

Sullivan, Michael (1996), Algebra and Trigonometry, Prentice Hall, p. 381, ISBN 0-13-370149-2.
Sehgal, V K; Gupta, Sonal, Longman ICSE Mathematics Class 10, Dorling Kindersley (India), p. 119, ISBN 978-81-317-2816-1.

A very important consequence of Theorem 3 is that the condition np = 1 is equivalent to saying that the Sylow p-subgroup of G is a normal subgroup (there are groups that have normal subgroups but no normal Sylow subgroups, such as S4).

Less trivial applications of the Sylow theorems include the focal subgroup theorem, which studies the control a Sylow p-subgroup of the derived subgroup has on the structure of the entire group. This control is exploited at several stages of the classification of finite simple groups, and for instance defines the case divisions used in the Alperin–Brauer–Gorenstein theorem classifying finite simple groups whose Sylow 2-subgroup is a quasi-dihedral group. These rely on J. L. Alperin's strengthening of the conjugacy portion of Sylow's theorem to control what sorts of elements are used in the conjugation.

In permutation groups, it has been proven in (Kantor 1985a, 1985b, 1990; Kantor & Taylor 1988) that a Sylow p-subgroup and its normalizer can be found in polynomial time of the input (the degree of the group times the number of generators). These algorithms are described in textbook form in (Seress 2003), and are now becoming practical as the constructive recognition of finite simple groups becomes a reality. In particular, versions of this algorithm are used in the Magma computer algebra system.

Mathematics Encyclopedia