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Polynomial Functions

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In mathematics, we often come across with polynomial functions. In fact, they are everywhere in math. A function that is defined in terms of a polynomial is said to be a polynomial function. It involves only non-negative powers of independent variable $x$. It can be quadratic, cubic, or quartic and so on. In this unit, we are going to discuss about the polynomial functions and various concepts based on them.

Definition

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A polynomial function can be defined as an equation in two variables. The left side of the equation is $f(x)$ which can be written $y$, while the right side is made up of polynomial expression having finite terms of positive integer powers of $x$. The general form of a polynomial function is:

This is usually expressed as

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

Some facts about polynomial functions are

(1) As we can see that the polynomial functions are represented in descending powers of their variables.

(2) The real numbers or $(-\infty,\ +\infty)$ is the domain of the polynomial functions.

(3) The coefficient of the highest power variable is known as the leading coefficient.

(4) The highest power among all the exponents in a polynomial function, is termed as the degree of that polynomial function.

Factors and Roots of Polynomial Functions

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When we simplify the polynomial functions, we obtain expressions of the form :

$(x\ -\ a_{1})\ .\ (x\ -\ a_{2})$ ... $(x\ -\ a_{n})$, where each $(x\ -\ a_{i})$ is said to the factor of given polynomial function. Also, each a$_{i}$ is called root or zero of the polynomial.

In other words, an expression $(x - a)$ is known as a factor of polynomial function $f(x)$, if and only if $f(a)$ = $0$. It means that if we substitute $x$ = $a$ in $f(x)$, we end up getting zero. Thus, the value "$x$ = $a$" is referred as a zero or root of the polynomial.

For example

$f(x)$ = $x^{2}\ -\ 3x$

$f(x)$ = $x\ (x - 3)$

So, $x$ and $x\ -\ 3$ are two factors of polynomial function $f(x)$.

and $x$ = $0$ and $x$ = $3$ are two roots or zeros of the given polynomial.

Graphing

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When we graph a polynomial function, we always obtain a smooth and continuous curve. The polynomial function graphs are free from holes, cusps, break or corners. They can have peaks (highs) and valleys (lows).

A general graph of a polynomial function may look like the following.

Polynomial Graph

On the other hand, following two graphs DO NOT demonstrate polynomial function graphs.

Discontinuous graph

and

Polynomial Graph With Corner and cusp

End Behaviors of The Graphs of Polynomial Functions

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The end behavior of a polynomial indicates the appearance of the graph of a polynomial function. It shows the behavior of the graph at its ends. The end behavior tells about the direction where endpoints of the graphs point; whether they point up or down.

There are following 4 cases according to which the end behavior of polynomial function is predicted.

Case 1: Degree Even and Leading Coefficient Positive

When degree of the polynomial is even and leading coefficient is positive, both endpoints of the polynomial curve point in upward direction, as shown in the following diagram.

Graphs End Behaviour

Case 2:
 Degree Even and Leading Coefficient negative

If the degree of the polynomial is even and leading coefficient is negative, then both endpoints of the polynomial curve point in downward direction. It is demonstrated in the graph below.

Graphs End Behaviour Example

Case 3:
 Degree Odd and Leading Coefficient Positive

If the degree of given polynomial is odd and leading coefficient is positive, then right endpoint points upwards and left endpoint points downwards, as shown as under :

Polynomial Graphs End Behaviour

Case 4:
 Degree Odd and Leading Coefficient Negative

If the degree of given polynomial is odd and leading coefficient is negative, then right endpoint points downwards and left endpoint points upwards which is illustrated below.

Example on Graphs End Behaviour

Examples

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Few examples based on polynomial functions are given below.
Example 1: 

Find if $x$ = $0$, -$1$ and -$3$ are the roots of the polynomial function $f(x)$ = $2x^{4}\ +\ 13x^{3}\ +\ 29x^{2}\ +\ 27x\ +\ 9\ ?$

Solution: 

$f(x)$ = $2x^{4}\ +\ 13x^{3}\ +\ 29x^{2}\ +\ 27x\ +\ 9$

Substituting $x$ = $0$

$f(0)$ = $2(0)^{4}\ +\ 13(0)^{3}\ +\ 29(0)^{2}\ +\ 27(0)\ +\ 9$

$f(0)$ = $9$

Therefore, $x$ = $0$ is NOT a root of $f(x)$

Substituting $x$ = -$1$

$f(-1)$ = $2(-1)^{4}\ +\ 13(-1)^{3}\ +\ 29(-1)^{2}\ +\ 27(-1)\ +\ 9$

$f(-1)$ = $2\ -\ 13\ +\ 29\ -\ 27\ +\ 9$ = $40\ -\ 40$

$f(-1)$ = $0$

So, $x$ = -$1$ is a root of $f(x)$

Substituting $x$ = -$3$

$f(-3)$ = $2(-3)^{4}\ +\ 13(-3)^{3}\ +\ 29(-3)^{2}\ +\ 27(-3)\ +\ 9$

$f(-3)$ = $2\ .\ 81\ -\ 13\ .\ 27\ +\ 29\ .\ 9\ -\ 27.3\ +\ 9$

$f(-3)$ = $162\ -\ 351\ +\ 261\ -\ 81\ +\ 9$

$f(-3)$ = $432\ -\ 432$

$f(-3)$ = $0$

Hence, $x$ = -$3$ is a root of $f(x)$.
Example 2: 

Determine the end behaviors of the following polynomials.

(i) -$7x^{3}\ +\ 4x^{2}\ -\ 6x\ +\ 11$

(ii) $4x^{2}\ -\ 7x\ +\ 3$

Solution: 

(i) -$7x^{3}\ +\ 4x^{2}\ -\ 6x\ +\ 11$

Leading coefficient = -$7$ = Negative

Degree = $3$ = Odd

Hence, the right endpoint of its graph will point in downward direction, while the left endpoint will point in upward direction.

(ii) $4x^{2}\ -\ 7x\ +\ 3$

Leading coefficient = $4$ = Positive

Degree = $2$ = Even

Hence, both the endpoints of given polynomial function will point in upward direction.