Show that the polynomial has a zero between – 2 and - 1.
X4 - 6x3 - 30x2 + 48x + 82

So, the polynomial has a zero between – 2 and - 1.

Step-by-step explanation:

To find the zeros (or roots) of the polynomial functions, we need to use the remainder theorem and intermediate value theorem.

So here also we will find the value of the polynomial at x=-1 and at x=-2.

So we get:

So here we see that the value of the polynomial at two different values of x, is opposite in sign. So this implies that there must be a zero between – 2 and - 1.

  b.   the average rate of change of f(x) is less than the average rate of change of g(x).

step-by-step explanation:

since the length of the interval is the same in each case, we only need to compare the change in y-value to see which is greater.

so the change in y-value for function f is (6 )) = 9.

the table tells us

so the change in y-value for function g is (31 -1) = 30.

the change in y-value of f(x) on the interval is less than the change in y-value of g(x) on the same interval, so the average rate of change of f(x) is less than the average rate of change of g(x).