# Polynomial Function

## Polynomial Function

Polynomial Function is defined as the real valued function $f : R \rightarrow R$ if for each $x \in R$ $y = f(x)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+.....+a_1x+a_0$
Where $a_n,a_{n-1},....,a_1,a_0$ are constant and real numbers and n is a non-negative integer
Some important points to remember
• an,an-1,...,a,a0 are called the coefficients for xn,xn-1 ,..,x,x0
• n is called the degree of the polynomial function
• A constant Function is the polynomial function with zero degree
• n is a non-negative integer.It can not be fraction also
Example of Polynomial functions
$3x^2 + 4x+6$
$9x^3 -4x^2 +1$
$4x +1$
$4x^2 + \sqrt {2}$

The below are not polynomial functions
$x^{-4} + 1$
$\sqrt {x} + 2x +1$
$x^{2/3} +1$

### Types of Polynomial function based on degree

 Type of Polynomial function Example Degree Constant function $1$ n=0 Linear function $4x +1$ n=1 Quadratic function $x^2 +4x +1$ n=2 Cubic Function $3x^3 -4x^2+x +11$ n=3 Quartic Function $x^4 +2x^3 +4x^2+x +1$ n=4

Function f(x) =0 is also an polynomial function with undefined degree

## Domain and Range of the Polynomial Function

Domain = R
Range is dependent on the type of polynomial function.
For Linear function ,Range is R
For constant function Range is {c}
For Quadratic function like ${x^2 +1}$ , Range is $[1,\infty)$

## Graph of the Polynomials Function

The graph of the Polynomial function depends on the type of polynomial functions.

### For constant and Linear function

It is always straight line for constant and Linear Function
Here are the graph for function
$f(x)=3$ ( Constant function)
$f(x)=x+2$(Linear Function)
$f(x)=3x+1$(Linear Function)

For Quadratic function, the graph is a parabolic graph. Lets see few example

1) $f(x) =2x^2$ , $h(x) =5x^2$ , $g(x) =10x^2$

We can see the graph is a upward parabola and as we increase the coefficient of $x^2$,the graph is stretched vertically

2) $f(x) =-2x^2$ , $h(x) =-5x^2$ , $g(x) =-10x^2$

We can see the graph is a downward parabola and as we increase the coefficient of $x^2$,the graph is stretched vertically

3) $f(x) = x^2 + 2x +1$ , $g(x) = x^2 + 4x +1$ ,$h(x) =x^2 +6x +1$

We can see the graph is a upward parabola and with increase in coefficient of x ,it drifts downwards on the left

4. $f(x) =x^2 +1$ and $h(x) =-x^2 -1$

Here since their are no real roots of the given quadratic function , the graph is not touching the x-axis. First graph is upward above x-axis and second graph is downward below x-axis
So , the intersection of parabola graph depends on the real roots of the function

5.$f(x) =x^2 -5x +6$ and $h(x) =-x^2 +7x -12$

Here since their are real roots of the given quadratic function ,the graph is intersecting the the x-axis.

### For Cubic function

Lets see few example of the cubic function
1. $f(x) =x^3$

We can see it extends in both upward and downward direction

2. $f(x) = x^3 + 1$

Here is the cubic function with one real root

3.$f(x) = x^3 -3x^2 +3x -1= (x-1)^2$

Here is the cubic function with three equal real root

4.$f(x) = x^3+12x^2+39x+28=(x+1)(x+4)(x+7)$

Here is the cubic function with three different real root.

### For Quartic function

1. $f(x) =x^4 -5x^2 +4 =(x-1)(x-2)(x+1)(x+2)$

This is a quartic function and it has four real roots .

Some important points on Polynomial functions graphs

1. The turning point in a graph is defined as the points from where graph from upward to downward or downward to upward. The turning points in the graph is always less or equal to (n-1) of the polynomial function.So a quartic function has maximum 3 turning points in the graph.A quadratic equation has maximum one turning point. A Cubic equation has maximum 2 turns. You can validate these in above given example graph also

2. The zero's or root of the polynomial function are point at which graph intersect x-axis , i,e the point where the value of y=0. The real roots of the polynomial function is always less or equal to the degree n of the polynomial.

## Solved examples of Polynomial Functions

1. Find the domain of the below function?
a. $y =x^2 +5$
b. $y = x^3$
c. $y=x^2 -7x +5$ Solution

a. Clearly it is defined for all $x \in R$ b. Clearly it is defined for all $x \in R$ c. Clearly it is defined for all $x \in R$ 2. Find the domain and range of function
$y= \frac {1}{\sqrt {x- [x]}}$
Solution
We have ,
$y= \frac {1}{\sqrt {x- [x]}}$
Domain of y
We know that
$0 \leq x -[x] < 1 , x \in R$
for $x \in Z ,x -[x]=0$
So Domain is R - Z
Range of y
$0 < x- [x] < 1 , x \in R -Z$
$0 < \sqrt {x -[x]} < 1$
$1 < \frac {1}{\sqrt {x- [x]}} < \infty$
So Range is = $(1,\infty)$

3. Let $f : R \rightarrow R$ , $f(x) = x^2 + 2[x] -1$ for each $x \in R$
Find the values of f(x) at x= 1.2 ,-.5 ,-2.1
Solution
$f(x) = x^2 + 2[x] -1$
$f(1.2) = (1.2)^2 + 2[1.2] -1 =1.44 +2-1=2.44$
$f(-.5) = (-.5)^2 + 2[-.5] -1 =.25 -2-1=-2.75$
$f(-2.1) = (-2.1)^2 + 2[-2.1] -1 =4.41 -6-1=-2.59$

### Quiz Time

Question 1Find the domain of the function $f(x)= \frac {1}{\sqrt {[x]-x}}$
A. R - Z
B. R
C. {}
D. Z
Question 2Find the range of values of x for which $[x]^2 -4 =0$ $A.$[-2,-1) \cup [2,3)$B.$[-2,-1] \cup [2,3]$C.$[-2,-3) \cup [2,3)$D.$[-2,-3] \cup [2,3)$Question 3 if$f(x) =[x]^2 +[x] +2$,then which of them is incorrect A. f(1)=4 B. f(.75)=2 C. f(-.50)=2 D. f(0) =3 Question 4 Value of the expression [{x}] + {[y]} is A. 1 B. 0 C. 2 D. Cannot be determined Question 5 The value {-.75} where {.} is the fractional part is A. -.75 B. .25 C. 0 D. None of these Question 6The domain of the function f given by$f (x) = \frac {1}{\sqrt {x^2 -[x]^2}}\$ is
A. Domain = R
B. Domain = R+
C. Domain = R+ -Z
D. Domain = Z