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 ...

, 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 t ...

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 exampl ...

. 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 ...

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 exampl ...

. 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 bic ...

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, bec ...

or synthetic division. # Conclude that any root

x \neq a

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

. 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, ...

, 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