k is a degree 5 polynomial that passes through the origin, has zeros i and 5 − i , and k ( − 3 ) = − 5850 . Find the equation for k ,

Answer 1

I suppose that polinomial k(x) has real coeffisients. Then it has zeros: 5 - i, 5 + i, i, - i and 0, so

k(x) = ax(x - i)(x + i)(x - 5 + i)(x - 5 - i) = ax(x² - i²)((x - 5)² - i²) = ax(x² + 1)(x² - 10x + 26);

k(- 3) = -3(9 + 1)(9 + 30 + 26) = - 5850; a = 3

So, k(x) = 3x(x² + 10)(x² - 10x + 26)