Question 970036

Form a polynomial {{{f(x)}}} with real coefficients having the given degree and zeros.

Degree {{{4}}}; 

zeros: 
{{{x[1]=2}}}, multiplicity {{{2}}}; =>use it two times, or square it
{{{x[2]=3i}}}=>we also have {{{x[3]=-3i}}} because complex zeros come always in pairs

then, using zero product rule, {{{f(x)}}} will be:

{{{f(x)=(x-x[1])^2(x-x[2])(x-x[3])}}}

{{{f(x)=(x-2)^2(x-3i)(x-(-3i))}}}

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

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

{{{f(x)=(x^2-4x+4)(x^2-9i^2)}}}

{{{f(x)=(x^2-4x+4)(x^2-9(-1))}}}

{{{f(x)=(x^2-4x+4)(x^2+9)}}}

{{{f(x)=x^4-4x^3+4x^2+9x^2-36x+36}}}

{{{f(x)=x^4-4x^3+13x^2-36x+36}}}


{{{ graph( 600, 600, -10,10, -10, 50, x^4-4x^3+13x^2-36x+36) }}}