[IB-HL H&H_8G Q16] Sum of Squares of some Binomial Coefficients

Question

Introduction

     This problem is taken from the Haese textbook for International Baccalaureate, 3^rd Edition, page 262.  It looks pretty daunting doesn’t it?  Where do we even begin?  The key to solving this problem is to realise that the binomial coefficients are coefficients of (numbers attached to) certain powers of  x  in the expansion.  The question is:  which power or powers?

     Before we go into that, let us review some important relevant facts.

Reminders

Solution

Final Remarks

     This problem was solved by using the symmetry property and treating binomial coefficients as coefficients of certain powers of  x.  We also worked backwards by noting that the RHS of the equation to be proven is the coefficient of  xⁿ.  This suggests that we compare this with the coefficients of  xⁿ  on the LHS.

HeuristicsUsed

H03. Make a systematic list

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* International Baccalaureate Mathematics (HL)

* GCE ‘A’ Levels H2 Mathematics

* other syllabuses that involve complex numbers and polynomials

[AM_20150413RSD] Rationalising Denominators for #Surds

Question 

Introduction

     Expressions with surds in their denominators are cumbersome.  The good news is that we can make the denominators into rational numbers, which are nicer.  Rational numbers those that can be expressed as a ratio of integers i.e. they are (proper or improper) fractions or can be converted to fractions.  Whole numbers are also part of rational numbers because you can always put them upon a denominator of  1;  e.g. 2 = ²/₁,  so  2  is a rational number.

     The standard trick for simplifying expressions with surds in their denominators is to rationalise the denominator by mutiplying the numerator and the denominator with its conjugate surd.  For example, the conjugate surd of   Ö5 + Ö2   is   Ö5 – Ö2.   Just change the  +  to  –  or the  –  to  +.  Let us see how the magic works.

Solution

Remarks

     Note that in the first step, I pulled out 2 as the common factor of the denominator, so that I get a simpler surd to work with.  Always try to work with simpler expressions.  This not only shortens your working, it reduces your chances of making a careless mistake.
     In mathematics, “rationalising” does not mean you give some reason or excuse for something that you know you have done wrong.  It means “make it into a rational number”.  Why does rationalising the denominator work?  This is because on the bottom (denominator) we have a difference-of-squares expression of the form
                                           (a + b)(a – b)   which is equal to   a² – b².
Since squaring “gets rid” of square roots,  a²  and  b²  will give you rational numbers (whole numbers or fractions), you will end up with a nice number downstairs (on the denominator).  Pupils should make sure they have this technique in their repertoire of skills.

Suitable Levels
* GCE ‘O’ Level Additional Mathematics
* revision for GCE ‘A’ Level H2 Mathematics
* revision for IB Mathematics HL / SL
* other syllabuses that involve surds
* precocious kids who always want to learn more

[AM_20150409ASD] Absurd Surds!

Question

Introduction

     This looks like a dastardly absurd problem on surds.  If you tried to do this question by direct calculation, you might get the correct answer after a long series of working, provided you do not make careless mistakes.

     Is there a better way to do it?  You bet!  First we simplify  x  as much as possible.  [H10]  Secondly, look for relationships or patterns.  [H04]  Note that  y  is the “inverted” version of  x, so  y  is the reciprocal of  x.  Thirdly, use more algebra [H13] instead of just calculating like a donkey.  Some useful algebraic identities are  x³ + y³ = (x + y)(x² – xy + y²),  and   x² + y² = (x + y)² – 2xy.

Solution

Heuristics Used

H04. Look for pattern(s)

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
* anyone who is interested in surds, even if it seems absurd.