Question 997792
given:
the polynomial of degree {{{5}}}, 
{{{P(x)}}} has leading coefficient {{{1}}}, 
has roots of multiplicity {{{2}}} at {{{x[1]=x[2]=2}}} and {{{x[3]=x[4]=0}}}, and a root of multiplicity {{{1}}} at {{{x[5]=-3}}} 

 a possible formula for {{{P(x)}}} is:

{{{P(x)=(x-x[1])(x-x[2])(x-x[3])(x-x[4])(x-x[5])}}}...since {{{x[1]=x[2]=2}}} and {{{x[3]=x[4]=0}}}, we can write it like this

{{{P(x)=(x-x[1])^2(x-x[3])^2(x-x[5])}}}....plug in given values

{{{P(x)=(x-2)^2(x-0)^2(x-(-3))}}}

{{{P(x)=(x^2-4x+4)(x^2)(x+3)}}}

{{{P(x)=(x^4-4x^3+4x^2)(x+3)}}}

{{{P(x)=x^5-4x^4+4x^3+3x^4-12x^3+12x^2}}}

{{{P(x)=x^5-x^4-8x^3+12x^2}}}


{{{drawing( 600, 600, -5, 5, -5, 50,
circle(2,0,.05),circle(0,0,.05),circle(-3,0,.05),
locate(2.2,1.4,p(2,0)),locate(0.2,1.4,p(0,0)),locate(-3,1.4,p(-3,0)),
 graph( 600, 600, -5, 5, -5, 50, x^5-x^4-8x^3+12x^2)) }}}