Polynomial function

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

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

where $n$ is a integer, and $a_{n},a_{n-1},a_{n-2},...,a_{0},\,$

are real numbers, and $a_{n}=O\,$

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 quotient 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 polynomiall 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:

Finding the value using Synthetic Division

By Synthetic Division

Finding the value using Remainder Theorem

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.

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 $P(x)=x^{3}-x^{2}-4x+4\,$ is divided by $x-2\,$ , the remainder is 0; that is, $P(2)=0\,$ .

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 $P(x)=(x-c)\bullet Q(x)\,$ . Since the equation is an identity and is true for any value of x, the it must be true for x = c

Then,

$P(c)=(c-c)\bullet Q(c)\,$

$P(x)=0\bullet Q(c)=0\,$

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,

$P(c)=(x-c)\bullet Q(x)+0\,$

$P(c)=(x-c)\bullet Q(x)\,$

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

External links