SOLUTION: For the polynomial Function : F(x)=-2(x + 1/2) (x+4)^3 a)List each real zero and its multiplicity: b)Determine whether graph crosses or touches the x-axis at each x-intercept: c

Question 1189459: For the polynomial Function : F(x)=-2(x + 1/2) (x+4)^3
a)List each real zero and its multiplicity:
b)Determine whether graph crosses or touches the x-axis at each x-intercept:
c)Determine the behavior of the graph near each x-intercept(zero):
d)Determine the maximum number of turning point on the graph:
e)Determine the end behavior, that is finding the power function that the graph of
f resembles for large values of |x|:

Found 2 solutions by MathLover1, Boreal:
Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!

For the polynomial Function :

a)List each real zero and its multiplicity:

if =>,..... multiplicity 1
if =>,..... multiplicity 3
b)Determine whether graph crosses or touches the x-axis at each x-intercept:

For zeros with multiplicities, the graphs or are tangent to the x-axis at these x-values.
For zeros with multiplicities, the graphs or intersect the x-axis at these x-values.
have an multiplicity => the graph will the x-axis
have an multiplicity=> the graph will the x-axis

c)Determine the behavior of the graph near each x-intercept(zero):
Since the leading term of the polynomial (the term in the polynomial which contains the highest power of the variable) is , the degree is , i.e. even, and the leading coefficient is , i.e. negative.
This means that f(x)→ -∞ as x→ -∞ , f(x)→ -∞ as x→ ∞

d)Determine the maximum number of turning point on the graph:
The maximum number of turning points of a polynomial function is always one less than the degree of the function.
This function f is a th degree polynomial function and has turning points.

e)Determine the end behavior, that is finding the power function that the graph of
f resembles for large values of ||:
If we expand we get

..... the dominating term

Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
Graph this:
;
By inspection, x=-4 is a root with multiplicity 3, and so is x=-0.5. Both of those make the parentheses 0, and they cross the x-axis.
-
Take the derivative of the function:
f'(x)=2(x+0.5)*3(x+4)^2+(x+4)^3(2)
as x approaches -4 from the either side, the derivative is negative, or the slope negative.
As x approaches 0.5 from either side, the slope is positive.
There is one turning point where the derivative is 0.
2(x+0.5)*3(x+4)^2+2(x+4)^3=0
divide both sides by (x+4)^2
6x+3+2(x+4)=0
8x+11=0
x=-1.375
y=-31.65
-
end behavior negative is (-x)^4 positive oo
and for positive is x^4 positive oo

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