Factor theorem
Encyclopedia
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
.
The factor theorem states that a polynomial
has a factor 
if and only if
.
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:
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 equation
.
and therefore
and 
are the factors of 
be a polynomial with complex coefficients, and 
be in an integral domain (e.g. 
). Then 
if and only if 
can be written in the form 
where 
is also a polynomial. 
is determined uniquely.
This indicates that those
for which 
are precisely the roots of 
. Repeated roots can be found by application of the theorem to the quotient 
, which may be found by polynomial long division
.
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, the factor theorem is a theorem linking factors and zeros
Zero (complex analysis)
In complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written asf=g\,where g is a holomorphic function g such that g is not zero.Generally, the...
of a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
. It is a special case
Special case
In logic, especially as applied in mathematics, concept A is a special case or specialization of concept B precisely if every instance of A is also an instance of B, or equivalently, B is a generalization of A. For example, all circles are ellipses ; therefore the circle is a special case of the...
of the polynomial remainder theorem
Polynomial remainder theorem
In algebra, the polynomial remainder theorem or little Bézout's theorem is an application of polynomial long division. It states that the remainder of a polynomial f\, divided by a linear divisor x-a\, is equal to f \,.- Example :...
.
The factor theorem states that a polynomial
has a factor if and only ifIf and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
.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 math 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 divisionPolynomial long divisionIn algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long 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 
.
Example
You wish to 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 equationQuadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
.
and therefore
and are the factors of Formal version
Let be a polynomial with complex coefficients, and be in an integral domain (e.g. ). Then if and only if can be written in the form where is also a polynomial. is determined uniquely.This indicates that those
for which are precisely the roots of . Repeated roots can be found by application of the theorem to the quotient , which may be found by polynomial long divisionPolynomial long division
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division...
.