Question

Find an nth degree polynomial function with real coefficients satisfying the given conditions..
N=3; -4 and 6+5i are zeros; f(1)=250
(if you know how to do this but dont feel like doing it for me; Could you please just inform me on how i's work while factoring, i.e. What do they affect, etc.

Answer 1

we know that if we have a polynomial with real coefients and one root is a+bi, another root is a-bi

so

some roots are

3,-4,6+5i, 6-5i

for roots, r1 and r2

teh factors are

(x-r1)(x-r2)

so

we can put them in and

(x-3)(x+4)(x-6-5i)(x-6+5i)

(x^2+x-12)(2x^2-12x+61)

x^4-11x^3+37x^2+205x-732

to keep the same coefients, we multiply whole thing by c

c(x^4-11x^3+37x^2+205x-732)

we don't know what c is

we know that

f(1)=250

f(1)=c(x^4-11x^3+37x^2+205x-732)

find c

f(1)=c(1^4-11(1^3)+37(1^2)+205(1)-732)=250

f(1)=c(-500)=250

-500c=250

divide both sides by -500

c=-1/2

the polynomial (factored with real coefients) is

f(x)=-0.5(x-3)(x+4)(2x^2-12x+61)