Question

Which statements are true about the graphs of all nth degree polynomials?

Choose all that apply.
A.) It must go up and down a total of n times.
B.) It goes up and down at most a total of n times.
C.) The number of x-intercepts is n.
D.) The number of x-intercepts is at most n.
E.) The end of the graph must go up either to the left or the right.
F.) The end behavior depends on the number of terms of the polynomial.

Answer 1

The statement that is true about the graphs of all nth degree polynomials is:

B.) It goes up and down at most a total of n times.

D.) The number of x-intercepts is at most n.

Explanation:

We know that the number of times the graph goes up and down depends on the number of distinct zeros of a polynomial and as we know that a polynomial of nth degree may have repetitive zero.

Hence, the graph of nth degree polynomial goes up and down at most a total of n times.

Also, the number of x-intercept is the x-value of the point where the function is zero i.e. it depends on the number of zeros of polynomial.

Hence, The number of x-intercepts is at most n.

Also, the end behavior of graph depends on degree as well as sign of the leading coefficient.

Hence, correct options are:

Option B and option: D

Answer 2

The statements are true about the graphs of all nth degree polynomials are B and D.

B. It goes up and down at most a total of n times.

D. The number of x-intercepts is at most n.