Jump to content

Polynomial function

From Wikipedia, the free encyclopedia
This is an old revision of this page, as edited by Johnreyrayala (talk | contribs) at 11:59, 18 January 2010. The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

Template:New unreviewed article

Polynomial functions are continuous. This means that the graph has no holes or gaps.

In mathematics, a polynomial function of degree is a function of the form:

where is an integer, and

are real numbers, and

The Remainder Theorem

If a polynomial P(c) is divided by x − c,where c is a real number,then the remainder is P(c).

Consider this division:

The quotient here is ,and the Remainder is 2.This result may also be expressed as . This means that there is a difference of 2 between the dividend ,and the product of the quotient and the divisor .

Division Algorithm For Polynomials

For each polynomial of the positive degree and any real number , there exist a unique polynomial and a real number such that:

where is a degree , and is the remainder.

Proof:

1.

2.

3.

4.

  • The Equation is true for all x,therefore. let .
  • Hence, The Remainder R is equal to P(c).

Finding Values of Polynomial Functions

Synthetic division , hand-in-hand with the Remainder Theorem can be used as a convenient way to find values of polynomial functions.

The Remainder Theorem states that when the polynomiall P(x) is divided by x − c, the remainder is P(c).

For examples,if the polynomial is divided by x − 3, the remainder is P(3).

Illustrative examples

A. Use synthetic division and the Remainder Theorem to Find The Value of at 3.

Solution:

Finding the value using Synthetic DivisionBy Synthetic Division Finding the value using Remainder TheoremBy Remainder Theorem

The Remainder is 27 .Therefore ,P(3)=27

In summary, the remainder R obtained in synthetic division of f(x) by x − c,provides these information:

  • The remainder R gives the value of f at x = c,that is, R = f(c).
  • If R = 0, then x − c is a factor of f(x).
  • If R = 0, then (c, 0) is an x intercept of the graph of f.

The Factor Theorem

Let P(x) be a polynomial. If P(c)= 0, where c is a real number, the x-c is a factor of P(x). Conversely, if x-c is a factor of P(x), then P(c)= 0

The Remainder Theorem states That when the polynomial P(x) is divided by x-c, the remainder is P(c).

For example, when is divided by , the remainder is 0; that is, .

The same result is obtained using synthetic division
File:Synthetic 2.jpg
Notice That P(c) in this case is 0. Since P(c) is 0,the equation becomes:
File:Factor 2.jpg

Since the theorem has a converse, the proof consists of two parts.

a. If (x-c) is a factor of P(x), then P(c) = 0 .

b. If P(c) = 0, then (x-c) is a factor of P(x).


Proof for a:

Suppose (x-c) is a factor of P(x),then . Since the equation is an identity and is true for any value of x, the it must be true for x = c

Then,


Proof for b:

Suppose P(c)= 0. By the Remainder Theorem, when P(x) is divided by (x-c), the remainder R = P(c)= 0.

Then,

Therefore,(x-c) is a factor of P(x)

References

  • Soledad Jose-Diloa,Ed.d.,Fernando B. Orines & Julieta G. Bernabe,"Advanced Algebra (Trigonometry and Statistics)" Textbook for Fourth Year ISBN 971-07-2227-1

See also