Polynomial function

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

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

$f(x)=a_{n}{x^{n}}+a_{n-1}{x^{n-1}}+a_{x-2}{x^{x-2}}+\cdots +a_{0}$

where n is a integer, and

$a_{n},a_{n-1},a_{n-2},\dots ,a_{0},$

are real numbers, and $a_{n}\neq 0.$

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:

${\frac {x^{3}-3x^{2}+x+4}{x-2}}=2\leftarrow remainder$

The qoutient here is $x^{2}-x-1\,$,and the remainder is 2.This result may also be expressed as $x^{2}-x-1+{\frac {2}{x-2}}$. This means that there is a difference of 2 between the dividend $x^{3}-3x^{2}+x+4\,$,and the product of the quotient $x^{2}-x-1\,$ and the divisor $x-2\,$.

Division algorithm for polynomials

For each polynomial $P(x)\,$ of the positive degree $n\,$ and any real number $c\,$ , there exist a unique polynomial $Q(x)\,$ and a real number $R\,$ such that:

$P(x)=(x-c)\cdot Q(x)+R\,$

where $Q(x)\,$ is a degree $n-1\,$ , and $R\,$ is the remainder.

Proof:

1.$P(x)=(x-c)\times Q(x)+R$

2.$P(c)=(c-c)\times Q(c)+R$

3.$P(c)=0\times Q(c)+R$

4.$P(c)=R\,$

• The Equation is true for all x,therefore. let $x=c\,$.
• 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 polynomial P(x) is divided by x − c, the remainder is P(c).

For examples,if the polynomial $P(x)=2x^{3}-8x^{2}+19x-12\,$ 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 $P(x)=2x^{3}-8x^{2}+19x-12\,$ at 3.

Solution:

By Synthetic Division By 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.

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

• Polynomial
• Function(Mathematics)
• Polynomial remainder theorem

External links

• Polynomial Functions
• Polynomial Functions