Geometrical Meaning of the Zeroes of a Polynomial

We know that  a real number k is a zero of the polynomial p(x) if p(k) = 0 in a geometrical way.

The x-coordinate of the point, where the graph of a polynomial intersect the x-axis is called the zero of the polynomial.

An nth - degree polynomial intersects the x-axis of n points and therefore, has a maximum of n zeros in geometrical graph

In a quadratic polynomial ax2 + bx +c,

If a>0, then the graph is a parabola that open upwards.

If a<0, then the graph is a parabola that open downwards.

Special case to the zeroes of a polynomial in geometrical meaning


In a geometrical way the evaluation of zeroes with polynomial evaluation of the equation in geometrical meaning is follows

Case (i) :

Here, the graph cuts x-axis at two distinct points A and A′.
The x-coordinates of A and A′ are the two zeroes of the quadratic polynomial ax2 + bx + c in this case is shown in the following figure



Case (ii) :

The graph given here cuts the x-axis at exactly one point, i.e., at two coincident points. So, the two points A and A′ of Case (i) coincide here to become one point A is shown in the following figure.



The x-coordinates  of the  A is the only zero for the quadratic polynomial ax2 + bx + c in this case.

No zero case

The graph given here is either completely above the x-axis or completely below the x-axis. So, it does not cut the x-axis at any point

So, the quadratic polynomial ax2 + bx + c has no zero in this case.
So, you can see geometrically that a quadratic polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero), or no zero. This also refers that the polynomial of degree 2 has at-most two zeroes is shown in the following figure.

Concept of the Derivative Online Help

In online, concept of derivative is clearly explained with the help of solved example problems. In online so many websites explain the concepts with the help of math sites. The derivative concept is clearly explained in calculus whereas derivative concept helps to find the rate of change for the given function with respect to change in the input function. The following are the example problems with detailed solution helps to explain the concept of derivative in online.


Concept of derivative online help example problems:


The following example problems explain the concept of derivative in online.

Example 1:

Determine the derivative by differentiating the polynomial function.

f(t) = 3t 3 +4 t 4  + 5t

Solution:

The given function is

f(t) = 3t 3 +4 t 4  + 5t

The above function is differentiated with respect to t to find the derivative

f '(t) = 3(3t 2 )+4(4t 3  ) + 5

By solving above terms

f '(t) = 9t 2 +  8t 3 + 5

Example 2:

Determine the derivative by differentiating the polynomial function.

f(t) = 6t6 + 5 t5 + 4 t4 + t

Solution:

The given equation is

f(t) = 6t6 + 5 t5 + 4 t4 + t

The above function is differentiated with respect to t to find the derivative

f '(t) =  6(6t 5)  +5 (5 t4 ) +4(4 t3) + 1

By solving above terms

f '(t) =  36t 5  + 25 t4  + 16 t3 + 1

Example 3:

Determine the derivative by differentiating the polynomial function.

f(t) = 2t 2 +4 t 4  + 15

Solution:

The given function is

f(t) = 2t 2 +4t 4  + 15

The above function is differentiated with respect to t to find the derivative

f '(t) = 2(2t  )+4(4 t 3 ) + 0

By solving above terms

f '(t) = 4t +16t3

Example 4:

Determine the derivative by differentiating the polynomial function.

f(t) = 5t5 +4t 4 +3t 3  + 2

Solution:

The given function is

f(t) = 5t5 +4t 4 +3t 3  + 2

The above function is differentiated with respect to t to find the derivative

f '(t) = 5(5t 4 )+4(4t 3 ) +3( 3t 2) +0

By solving above terms

f '(t) = 25t 4 +16t 3  +9 t 2


Concept of derivative online help practice problems:


1) Determine the derivative by differentiating the polynomial function.

f(t) = 2t 3 +3 t 4  + 4 t 5

Answer: f '(t) = 6t 2 +12 t3 + 20 t 4

2) Determine the derivative by differentiating the polynomial function.

f(t) = t 3+t5 + 4 t 6

Answer: f '(t) = 3t2 + 5t4 + 24 t 5