How do you find the polynomial given the following information?

Question

x=-3-5i, x=4+√6, and f(1)=246

When I tried to solve this problem, I got x^4-2x^3+2x^2-176x+544 but when I checked it, it was incorrect. Thank you for your help in advance!

Answer 1

x = -3±5i (remember complex numbers come in conjugates)

(x+3)² = -25

x²+6x+9 = -25

x²+6x+34 = 0

x = 4±√6 (also a conjugate pair here)

(x-4)² = 6

x²-8x+16 = 6

x²-8x+10 = 0

(x²+6x+34)(x²-8x+10) = x⁴-8x³+10x²+6x³-48x²+60x+34x²-272x+340 = x⁴-2x³-4x²-212x+340

f(x) = x⁴-2x³-4x²-212x+340

f(1) = 1⁴-2(1)³-4(1)²-212(1)+340 = 123

Since 246 is twice as big as 123, we can simply multiply the function by 2 and get our final answer:

f(x) = 2x⁴-4x³-8x²-424x+680

Hope this helped!