[AM_20150510PLRF27] Looking for a Polynomial's Missing Link

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  a₀  for the numerator and

all the possible factors of the coefficient  a_n  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.

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  v³,  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.

Heuristics Used

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)

* other syllabuses that involve polynomials, Remainder and Factor Theorem