algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

, the factor theorem is a

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

linking factors and zeros of a

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

. It is a

special case In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case is ...

of the

polynomial remainder theorem In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exam ...

. The factor theorem states that a polynomial

f(x)

has a factor

(x - \alpha)

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

f(\alpha)=0

(i.e.

\alpha

is a root).

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:. # Deduce the candidate of zero

a

of the polynomial

f

from its leading coefficient

a_n

and constant term

a_0

. (See

Rational Root Theorem In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or theorem) states a constraint on rational solutions of a polynomial equation :a_nx^n+a_x^+\cdots+a_0 = 0 with integer coefficients a_i\in ...

.) # Use the factor theorem to conclude that

(x-a)

is a factor of

f(x)

. # Compute the polynomial

g(x) = \frac

, for example using

polynomial long division In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, becaus ...

synthetic division In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division. It is mostly taught for division by linear monic polynomials (known as the Ruffini ...

. # Conclude that any root

x \neq a

f(x)=0

is a root of

g(x)=0

. Since the

polynomial degree In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus i ...

g

is one less than that of

f

, it is "simpler" to find the remaining zeros by studying

g

. Continuing the process until the polynomial

f

is factored completely, which its all factors is irreducible on

\mathbb /math> or \mathbb /math>.

Example

Find the factors of

x^3 + 7x^2 + 8x + 2.

Solution: Let

p(x)

be the above polynomial :Constant term = 2 : Coefficient of

x^3=1

All possible factors of 2 are

\pm 1

and

\pm 2

. Substituting

x=-1

, we get: :

(-1)^3 + 7(-1)^2 + 8(-1) + 2 = 0

So,

(x-(-1))

, i.e,

(x+1)

is a factor of

p(x)

. On dividing

p(x)

(x+1)

, we get : Quotient =

x^2 + 6x + 2

Hence,

p(x)=(x^2 + 6x + 2)(x+1)

Out of these, the quadratic factor can be further factored using the

quadratic formula In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, gr ...

, which gives as roots of the quadratic

-3\pm \sqrt.

Thus the three irreducible factors of the original polynomial are

x+1,

x-(-3+\sqrt),

and

x-(-3-\sqrt).

References

{{reflist Theorems about polynomials