First Principles AI

Solution

\begin{aligned} & \boxed{\text{Let } u = z^2:} \\ \\ & u^2 + 21u - 100 = 0 \\ \\ & \boxed{\text{Discriminant:}} \\ & \Delta = 21^2 - 4(1)(-100) = 441 + 400 = 841 \\ & \sqrt{\Delta} = 29 \\ \\ & \boxed{\text{Solve for } u:} \\ & u = \frac{-21 \pm 29}{2} \\ & u_1 = \frac{-21 + 29}{2} = \frac{8}{2} = 4, \quad u_2 = \frac{-21 - 29}{2} = \frac{-50}{2} = -25 \\ \\ & \boxed{\text{So:}} \\ & z^4 + 21z^2 - 100 = (z^2 - 4)(z^2 + 25) \\ \\ & \boxed{\text{Factor each:}} \\ & z^2 - 4 = (z - 2)(z + 2) \\ & z^2 + 25 = (z - 5i)(z + 5i) \\ \\ & \boxed{\text{Therefore:}} \\ & z^4 + 21z^2 - 100 = (z - 2)(z + 2)(z - 5i)(z + 5i) \end{aligned}