Question 1189459


For the polynomial Function : 

{{{f(x)=-2(x + 1/2) (x+4)^3}}}


a)List each real zero and its multiplicity:

{{{-2(x + 1/2) (x+4)^3=0}}}
if {{{(x + 1/2)=0}}}=>{{{x=-1/2}}},..... multiplicity 1
if {{{(x+4)^3=0}}}=>{{{x=-4}}},..... multiplicity 3

b)Determine whether graph crosses or touches the x-axis at each x-intercept:


For zeros with{{{ even}}} multiplicities, the graphs {{{touch}}} or are tangent to the x-axis at these x-values. 
For zeros with {{{odd}}} multiplicities, the graphs {{{cross}}} or intersect the x-axis at these x-values. 

{{{x=-4}}} have an {{{ even}}}  multiplicity => the graph will {{{touch}}} the x-axis 

{{{x=-1/2}}} have an {{{odd}}} multiplicity=> the graph will {{{cross}}}   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 {{{-2x^4}}}, the degree is {{{4}}}, i.e. even, and the leading coefficient is {{{-2}}}, 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 {{{4}}}th degree polynomial function and has {{{3}}} turning points. 


e)Determine the end behavior, that is finding the power function that the graph of
f resembles for large values of |{{{x}}}|:

If we expand we get
{{{f(x) = -2x^4 - 25x^3 - 108x^2 - 176x - 64}}}
{{{f(x) = -2x^4}}} .....  the dominating term