Question

A polynomial is to be constructed that has 8 turning points. what is the minimum degree of the polynomial?

a polynomial that has 5 turning points is multiplied by another polynomial that has 3 zeros. what is the minimum degree of the new polynomial?

a polynomial is to be constructed that will touch the x axis at most 5 times. what is the minimum degree of the polynomial?

given the following polynomial find the maximum possible number of turning points.
f(x)=x^14 x^6 x^9 x^3

i tried them and i got them all wrong can someone help me out?

Answer 1

1) 8 turning points means 8 vertices, which implies minimum degree 9 (when the polynomial is x^2 the degree is 2 and it has one turning point).

2) 5 turning points means minimum degree 6, 3 zeros means minimum degree 3, the multiplication implies minimum degree 6+3 = 9

3) 5 times crossing the x - axis is 5 zeros, then the minimum degree is 5.

4) f(x)=x^14 + x^6 + x^9 + x^3,maximum number of turning points 14 -1 = 3 (remember one less than the grade is the maximum posible number of turning points, but it could be less, for example f(x) = x^2 has one turning point).