Chapter 5.6, Problem 48PPSE

Solution

To find: The number of maximum points of intersection between the graphs of a quartic and a quantic polynomial function.

The maximum points of intersection are FIVE.

Given information: It is given that any odd degree polynomial equation with real coefficient is present.

Explanation:

Consider the equation $p(x)=q(x)$

The roots of this equation are the $x$ -coordinates of the points of intersection between the curves $y=p(x)$ and $y=q(x)$

Therefore, the number of points of intersection between the curves $y=p(x)$ and $y=q(x)$ are the number of roots of the equation $p(x)=q(x)$ .

Let $p(x)$ be a quantic polynomial and $q(x)$ be a quartic polynomial.

Then the equation $p(x)=q(x)$ will be a quantic polynomial.

The number of points of intersection between the curves representing the quantic and quartic polynomial are the number of roots of the equation $p(x)=q(x)$ .

By fundamental theorem of algebra, the maximum number of roots for a quantic polynomial is 5.

The maximum of points of intersection between the curves can be FIVE.

Example:

  $x^5+x^2+6x$ -Blue curve

  $x^4+7x^3$ -Black curve