Question

For the polynomial function below
a) List each real zero and its multiplicity.
b) Determine whether the graph crosses or touches the x-axis at each x-intercept.
c) Determine the behavior of the graph near each x-intercept.
d) Determine the maximum number of turning points on the graph.
e) Determine the end behavior; that is, find the power function that the graph of f resembles for large values absolute value of (x).

f(x)-7(x-7)(x+7)^2

Answer 1

a. There is only 1 real 0, at x = 5. The root x=5 has multiplicity 3.
This is all because (x2 + 1) has no real factor, and 5 is the root from (x-5)3 The root has multiplicity 3.
b. Since the real zero 5 has odd multiplicity, 3, the graph crosses the axis at this intercept.
c. Since it has multiplicity 3, though it crosses the axis, it does flatten out at this point.
d. While a fifth degree polynomial can have up to 4 turning points, a little reflection regarding this graph should lead us to the conclusion that it will have at most 2. In particular, the multiplicity 3 root eliminated 2 of the 4 possible relative maxima or minima. By turning point, I mean the existence of a relative maximum or minimum.
A graph is shown below. This verifies our speculation. If instead of (x2 + 1), we had x2, the turning points would be at 0 and 2 (2/5 of the way between 0 and 5), and the move of the turning points from 0 and 2 reflects the impact of the (+1) (Of course, if there was no constant, we would have another zero as well). Note that if it had been (x2 + 2), instead, we would have had no turns.

e. We get the end behavor by considering the leading coefficients. Thus, it behaves like -9x5

Answer 2

Does this help ? :) If so, click "best answer". ^_^

Answer 3

hell, man, im giving 2 medals just for writing all that !!

Answer 4

lol thx :)

Answer 5

so for a) the real zero will be 5 and multiplicity is 3, is it the high and low multiplicity?

Answer 6

where did you get the (x2+1)??? @dumbsearch2

Answer 7

didn't get my question answer but thanks