[H2_20151123APGP] Factor Theorem with Arithmetic and Geometric Progression

Question

Introduction

     This question tests students on their knowledge of arithmetic and geometric series.  They should also be familiar with Factor Theorem and methods of dealing with polynomials.  Once parts (i) and (ii) are solved, part (iii) is quite straightforward, provided that the student remembers how to deal with surds.

Review of Important Facts

Solution

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘A’ Level H2 Mathematics

* IB HL Mathematics

* other syllabuses that series and Factor Theorem

[AM_20151105QFER] New Quadratic Equation satisfied from New Roots

Question

The roots of the quadratic equation  2x2 – 3x + 6 = 0  are  a  and  b. (i)   Without finding the value of  a,  show that  8a4 = 18 – 45a. (ii)  Find the quadratic equation whose roots are  (a2 + 1)  and  (b 2 + 1).

Introduction

     Do you know what a “root” is?  Is it like radish or ginseng?  Do you know what “satisfied” means?  Is it that nice feeling you get when you eat carrots?  Read on!

     The featured problem above is modified from an original question that contained an error.  The modified part is shown in red.  I present two solutions.  The first solution uses pretty much standard theory, and I use notations  a’  and  b’  to denote the new roots  (a ² + 1)  and  (b ² + 1)  respectively.  For the second solution, I present an alternative working part (i), and one using the method of substitution for obtaining new equations (not usually taught in schools at the secondary level) for part (ii).  But before that let me first explain what “root” and “satisfied” means.

Recapitulation of Standard Theory

Solution 1 – Using Standard Theory

Solution 2 – Using the Method of Substitution for part (ii)

 Remarks

     Once again we can see that there are many ways to skin the cat, as it were.  Mathematics is not about following a fixed procedure.  There are various truths, notions and rules that are inviolable.  But other than that, you can have as much creativity as you want!

HeuristicsUsed

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* other syllabuses that involve quadratic roots

* whoever loves roots and enjoy being satisfied by conquering mathematical challenges J

[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