Question 1131984


 Find an equation for {{{f(x)}}}, the polynomial of smallest degree with real coefficients such that {{{f(x)}}} 


breaks through the x-axis at {{{-4}}},->point ({{{-4}}},{{{0}}})=> zero {{{x[1]}}}

if breaks through the x-axis at {{{5}}}, ->point ({{{5}}},{{{0}}})=> zero {{{x[2]}}}

has complex roots of:

 {{{x[3]=-3-i}}} => must also have {{{x[4]=-3+i}}} 

and {{{x[5]=5+5i}}} => must also have {{{x[6]=5-5i}}} 

and passes through the point ({{{0}}},{{{-54}}}) => zero {{{x[7]}}}



{{{f(x)=(x-x[1])(x-x[2])(x-x[3])(x-x[4])(x-x[5])(x-x[6])(x-x[7])}}}


{{{f(x)=(x-(-4))(x-5)(x-(-3-i))(x-(-3+i))(x-(5+5i))(x-(5-5i))(x-(-54))}}}


{{{f(x)=(x+4)(x-5)(x+3+i)(x+3-i)(x-5-5i)(x-5+5i)(x+54)}}}


{{{f(x)=(x - 5) (x + 4) (x + 54) (x^2 - 10 x + 50) (x^2 + 6 x + 10)}}}

:
:
{{{f(x)=x^7 + 49 x^6 - 286 x^5 - 584 x^4 + 15420 x^3 + 11700 x^2 - 253000 x - 540000}}}