Find an​ nth-degree polynomial function with real coefficients satisfying the given conditions.

Question

Find an​ nth-degree polynomial function with real coefficients satisfying the given conditions.

n= 3; 5, and 3-3i are zeros

Answer 1

Greetings! Lets solve this shall we ?

So, we have the factors such that x = 3, 5 and 3-3i. This implies a fourth zero such that 3 + 3i. Then the factors yield,

(x-3)(x-5)(x-(3-3i))(x+(3+3i)).................Now we distribute the factors together such that

(x-(3-3i))(x+(3+3i)) = x² + 6xi - 18 and

(x-3)(x-5) = x²-8x+15................Then,

(x²-8x+15)(x²+6xi-18)...............Now we distribute everything on the inside such that

x⁴ + 6x³i - 8x³ - 48x²i - 3x² + 90xi + 144x - 270.............Then simplify such that

x⁴ + x³(6i - 8) - 3x²(16i + 1) + 18x(5i + 8) - 270

I hope this helped!

Answer 2

it's a fourth degree polynomial with roots 3, 5, 3-3i and 3+3i because complex roots occur in conjugate pairs

(n-3)(n-5)(n-[3+3i])(n-[3-3i])

you can finish up the multiplication...