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:
- "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.)
- Use the factor theorem to conclude that is a factor of .
- Compute the polynomial , for example using polynomial long division or synthetic division.
- 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 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
Let be a one-variable polynomial with coefficients in a commutative ring , and let . Then if and only if for some polynomial . In this case, is determined uniquely.
As for the problem of algorithmically finding all roots, if is given and is known, then can be computed by polynomial long division; then one can compute the remaining roots of , including repeated roots, by factoring .
- 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.
- Bansal, R. K., Comprehensive Mathematics IX, Laxmi Publications, p. 142, ISBN 81-7008-629-9.