**Show that**

**can be written as**

**when Q(x) is a quadratic expression. Show that 2Q(x) can be written as the sum of three expressions, each of which is a perfect square.**

This is just the first part from Question 4 of the STEP1 2007 paper. It quickly becomes clear if one tries to use long division that (x+b+c) doesn’t provide a clean, easy division. The STEP instructions explain that the fastest route in answering the questions is always by inspection.

We know from the factor theorem that (x+b+c) is a factor, with no remainder:

Q(x) is therefore a quadratic expression in the following form:

The task is to determine whether each coefficient is positive or negative

The first equation has no coefficient of etc. so Q(x)(x+b+c) must form counteracting coefficients.

The only way to form this is

When multiplying out, this forms

So

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