All fortnightly questions

Despite mentioning complex numbers, you don't need to know a lot about them to solve this problem.  The main skills needed are algebraic manipulation and logical thinking.

Depending on what method you use, it might be helpful to know that the roots of polynomials with real coefficients are either real, or occur in complex conjugate pairs, and that a cubic with real coefficients always has at least one real root.

This STEP question does not require a lot of STEP knowledge, apart from factorising a linear factor from a quartic polynomial.  This video shows you one method for factorising a linear factor from a cubic polynomial and you can use a similar method with a quartic polynomial.

Carefully drawn diagrams can be helpful, as can considering specific cases.

This question is about rewriting functions in a different form.  No knowledge beyond GCSE is needed to complete the question.

The original STEP question had no question parts.  We have included some to make it clearer how the different requests in the question are related to each other.

This question involves substituting numbers into functions, simultaneous equations, and a bit of very basic number theory - for example, if $p$ and $q$ are both integers then so is $p+q$.

If you have not met it already, the formula for the triangular numbers is $\frac{n(n+1)}2$.  This is always an integer as we know that the triangular numbers are integers. Alternatively you could argue that $n$ and $n+1$ are consecutive numbers so one of them is even, and so has a factor of two.  Hence $\frac{n(n+1)}2$ must be an integer.