STEP 1 Question: A Step-by-step guide

Show that

Image

can be written as

Image

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:

Image

Image

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

Image

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

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

The only way to form this is

Image

When multiplying out, this forms

Image

Image

SoImage

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