Q. 100 - Precalculus Enhanced with Graphing Utilities

Question

A circle and a line intersect at most twice. A circle and a parabola intersect at most four times. Deduce that a circle and the graph of a polynomial of degree 3 intersect at most six times. What do you conjecture about a polynomial of degree 4? What about a polynomial of degree n? Can you explain your conclusions using an algebraic argument?

 

Step-by-Step Solution

Answer

We conjecture that a circle and graph of polynomial of degree n  intersect at most 2n times.   

Step 1. Given information

A circle and a line intersect at most twice. A circle and a parabola intersect at most four times.

Step 2. Explanation

As we know that number of solutions is equal to the product of the degrees of the equation. 

1. A circle and a line intersect at most twice. 

No. of intersection points = degree of circle x degree of line = 2x 1=2

2. . A circle and a parabola at most four times.  

No. of intersection points = degree of circle x degree of  parabola  =

 2x 2=4

3.  A circle and a polynomial of degree 3 intersect at most six times.   

No. of intersection points = degree of circle x degree of polynomial  = 2x 3=6

So, we conjecture that a circle and graph of polynomial of degree 4  intersect at most 8 times.

No. of intersection points = degree of circle x degree of polynomial  = 2x 4=8

So, we conjecture that a circle and graph of polynomial of degree n  intersect at most 2n times.

No. of intersection points = degree of circle x degree of polynomial  = 2xn=2n