Complex roots of equations

The function f is a quartic function with real coefficients.

The complex number 5i is a root of the equation f(x) = 0

Which one of the following must be a factor of f(x)?

Circle your answer.

(x² – 25)
(x² – 5)
(x² + 5)
(x² + 25)

Latifa and Sam are studying polynomial equations of degree greater than 2, with real coefficients and no repeated roots.

Latifa says that if such an equation has exactly one real root, it must be of degree 3

Sam says that this is not correct.

State, giving reasons, whether Latifa or Sam is right.

Find the other three roots.

Given that z 1 3i is one root of the equation z² pz r 0, where p and r are real, find the value of r.

Circle your answer.

Show that the equation

has exactly four solutions and state these solutions.

Plot the four solutions to the equation in part (a) on the Argand diagram below and join them together to form a quadrilateral with one line of symmetry.

Find the value of p and the value of r.

Abel and Bonnie are trying to solve this mathematical problem:

Abel says he has solved the problem.

Bonnie says there is not enough information to solve the problem.

Abel’s solution begins as follows:

State one extra piece of information about m and p which could be added to the problem to make the beginning of Abel’s solution correct.

Prove that Bonnie is right.