Question 1197185
a. 

{{{f(x) = - 7(x + 2)3(x - 1)2(x - 3) }}}

zeros/multiplicity /Crosses/Touches:  

zeros:
{{{0= - 7(x + 2)3(x - 1)2(x - 3) }} 
{{{0= (x + 2) }} =>{{{x=-2}}}
{{{0= (x -1) }} =>{{{x=1}}}

.{{{0= (x -3) }} =>{{{x=3}}}

multiplicity:

{{{x=-2}}} (Multiplicity of {{{1}}})

 {{{x=1}}} (Multiplicity of {{{1}}})
{{{x=3}}} (Multiplicity of {{{1}}})

Crosses/Touches: {{{3}}} Crosses

If a root of a polynomial has odd multiplicity, the graph will cross the {{{x}}}-axis at the the root. 

Degree: {{{3}}}
{{{f(x) = -42x^3 + 84x^2 + 210x - 252}}}

Sign: There is two sign change, so there is {{{2}}} positive root

End behavior:  

as {{{x ->-infinity}}}, {{{f(x)-> infinity}}}
as {{{x ->infinity}}}, {{{f(x)-> -infinity}}}


{{{ graph( 600, 600, -10, 10, -350, 350, -42x^3 + 84x^2 + 210x - 252) }}}



b. 

{{{m(x) = x^3 - 4x^2}}}

Factored form: {{{m(x) =x^2(x - 4)}}}

Degree Zeros Multiplicity Crosses/Touches:
{{{0=x^2}}}=>{{{x=0}}}(Multiplicity of {{{2}}})
{{{0=x-4}}}=>{{{x=4}}}(Multiplicity of {{{1}}})

Sign: no sign changes, so there are no negative roots

End behavior:
{{{x ->infinity}}}, {{{f(x)-> infinity}}}
{{{x ->-infinity}}}, {{{f(x)-> -infinity}}}


{{{ graph( 600, 600, -10, 10, -10, 10, x^2(x - 4)) }}}