#1. Make up a polynomial that has the following characteristics:
crosses the x axis at 1 and 4, touches the x axis at 0 ad 2. and is above the xaxis between 0 and 2.
I did (x+1)(x-4)(x^2)(x-2)^2
#2. its an illustration but one of the queestino says what is the minimum degree of polynomial. How do I find that and how do I find the maximum nuber of turnin points. I know it has to do with n-1 but Idont know how.
Thanks!!
should be (x-1)(x-4) to cross at 1 and 4
should be x^2 and (x-2)^2 to touch at 0 and 2
The last part confuses me. Since it crosses at x=1, how can it be positive for 0<x<2?
Or, on the other hand, since it crosses at x=1, it must be positive somewhere between 0 and 2.
Or, assuming your (x+1) factor is correct, then it shopuld have crossed at -1, not 1, and we have to consider how to make it positive for 0<x<2.
So, since we have
(x+1) x^2 (x-2)^2 (x-4)
for 0<x<2, all the factors except (x-4) are positive. So, try
-(x+1) x^2 (x-2)^2 (x-4)
To determine the minimum degree of a polynomial, you need to count the number of distinct x-intercepts, including those that touch the x-axis. In this case, the polynomial you provided has x-intercepts at 1 and 4. Therefore, the minimum degree of the polynomial is 2 (since the factors (x-1) and (x-4) account for those intercepts).
To find the maximum number of turning points (also known as the "degree minus one"), you can use the Exponent Rule. The exponent rule states that if a polynomial has a degree of n, then the maximum number of turning points is n-1.
In this case, since the degree of the polynomial is 2 (as determined above), the maximum number of turning points is 2-1 = 1.
So, the minimum degree of the polynomial is 2, and the maximum number of turning points is 1.
To find the minimum degree of a polynomial, you need to count the number of distinct x-intercepts or roots. In this case, the polynomial given in question #1 has two roots at x = 1 and x = 4. Therefore, the minimum degree of the polynomial should be 2 (since two distinct roots can be represented by a quadratic equation).
To find the maximum number of turning points, you can use the formula n - 1, where n represents the degree of the polynomial. From question #1, we determined the minimum degree to be 2. So, if a polynomial has a degree of 2, the maximum number of turning points it can have is 1 (2 - 1).
In summary, the polynomial from question #1 has a minimum degree of 2 and can have at most 1 turning point.