Question 1
If $$a$$, $$b$$ and $$c$$ are real numbers such that the polynomial $$x^3 + 6x^2 + ax + b$$ is the cube of $$x + c$$, then
Solution
Expanding gives $$(x+c)^3 = x^3 + 3cx^2 + 3c^2x + c^3$$. Comparing coefficients gives $$3c = 6$$, so $$c = 2$$, $$a = 3c^2 = 12$$ and $$b = c^3 = 8$$. Hence $$a+b+c = 22$$, which is divisible by $$11$$.
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