**Question**

**Introduction**

The first
part of this question is rather standard.
The second part is quite challenging, especially if you are trying to
connect with the earlier part.

**Reminder**

To
solve polynomial equations of degree
3 or higher (that are tested in
school tests and exams), we often need to guess a

*rational*(fraction or integer) root. By the way, whole numbers are rational numbers because we can always put them into fractions upon 1 as denominator. So how do we guess the roots? The following is a very important theorem that guides us as to what numbers to try.
So we consider all the possible factors of the
constant term

*a*_{0}for the numerator and
all the possible factors of the coefficient

*a*of the highest power for the denominator and consider the + and the – of all the possible fractions formed. Usually, we try those with denominator 1 i.e. the integers first._{n}**Solution**

**Remarks**

Note
that the solution consists of only the part in blue. Black is used for explanations, which are
lengthy because of the dense interplay of ideas and subtleties involved.

For the
second part, if you are not able to see the connection, then use the standard
method to solve the equation. Here we
realise that when the *x*is replaced by

*/*

^{v}_{2}, the coefficients are reduced to the original coefficients. However, these are in reverse order. This indicates that one needs to use the reciprocal, so you divide throughout by

*v*

^{3}, so that the highest power becomes just a constant. I know you would not have thought of this if you have not seen this kind of question before, but this is the trick to use.

Do not be
discouraged by difficult question. Have
a growth mindset. Every time you
encounter a difficult question, learn how the trick ticks. Your brain muscles will get stronger. Try to apply the same trick when you see a
similar question next time.

H04. Look for
pattern(s)

H05. Work
backwards

H09. Restate
the problem in another way

H10. Simplify
the problem

H11. Solve part
of the problem

H13* Use
Equation / write a Mathematical Sentence

**Suitable Levels**

*****GCE ‘O’ Level Additional Mathematics

*****GCE ‘A’ Level H2 Mathematics (revision)

*****IB Mathematics (revision)

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