Question 1176413
1.

{{{y=(x^2-5) (x-1)^2 (x-2)^2}}}....multiply and you will get
{{{y=x^6 - 6 x^5 + 8 x^4 + 18 x^3 - 61 x^2 + 60 x - 20}}}

A. Leading term  is {{{x^6}}}
B. X-intercept  are at x=sqrt(5), x=-sqrt(5),x=1,x=2
C. Y-intercept is ({{{0}}},{{{-20}}})
D. Multiplicity of roots:
{{{x=sqrt(5) }}} Multiplicity 1
{{{x= -sqrt(5)}}}  Multiplicity 1
{{{x=1}}} Multiplicity 2
{{{x=2}}}  Multiplicity 2
E. Number of turning points is {{{3}}}

note:
A polynomial of degree {{{n}}}, will have a maximum of {{{n -1}}} turning points.
The total number of turning points for a polynomial with an even degree is an odd number.
  example:  A polynomial with degree of {{{8}}} can have {{{7}}}, {{{5}}}, {{{3}}}, or{{{ 1}}} turning points
The total number of points for a polynomial with an odd degree is an even number.
   example:  A polynomial of degree {{{5}}} can have {{{4}}},{{{ 2}}}, {{{0}}} turning points (zero is an even number).


F. Sketch

{{{ graph( 600, 600, -5, 5, -5, 5, (x^2-5)*(x-1)^2*(x-2)^2,(x^2-5)*(x-1)^2*(x-2)^2) }}}


2.

{{{y=2x^4-3x^3-18x^2+6x+28}}}
{{{y = (x + 2) (2x - 7) (x^2 - 2)}}}

zeros: 
A. Leading term is {{{2x^4}}}
B. X-intercept are: 
({{{-2,0}}})
({{{7/2, {{{0}}})
({{{sqrt(2), {{{0}}})
({{{ -sqrt(2), {{{0}}})

C. Y-intercept is ({{{0}}},{{{28}}})
D. Multiplicity of roots  
 {{{x=-2}}} Multiplicity of 1
{{{x=7/2}}}, Multiplicity of 1
{{{x=sqrt(2)}}},Multiplicity of 1
{{{x=-sqrt(2)}}},Multiplicity of 1

E. Number of turning points {{{3}}}

F. Sketch

{{{ graph( 600, 600, -10, 10, -45, 35, 2x^4-3x^3-18x^2+6x+28,2x^4-3x^3-18x^2+6x+28) }}}