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Consider the polynomial P(z) ≡ z4 - 6z3 - 2z2 + 58z - 51 , z ∈ . (a) Express P(z) in the form (z2 + az + b)(z2 + cz + d) where a , b , c , d ∈ .

(c) Hence, or otherwise, state the condition on k ∈ such that all roots of the equation P(z) = k are real.

Steffi the stray cat often visits Will’s house in search of food. Let X be the discrete random variable “the number of times per day that Steffi visits Will’s house”. The random variable X can be modelled by a Poisson distribution with mean 2.1. (a) Find the probability that on a randomly selected day, Steffi does not visit Will’s house

b) Copy and complete the probability distribution table for Y . [4] y 0 1 2 3 4 P(Y = y)

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