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 only way to form this is
When multiplying out, this forms