[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

[IB-HL H&H_Rev6C Q11] Factors of a Complex Polynomial

Question

Introduction

     This question is taken from the Haese & Harris textbook for IB HL Mathematics and is rather challenging.  Here I present two solutions.  In the first solution, I use substitution to make a variable “disappear”.  [ heuristics: H10. Simplify the problem, H11. Solve part of the problem]  And that allowed me to crack the rest of them problem.  In the second solution, I rephrased the problem in terms of tangents to curves. [ heuristic H09. Restate the problem in another way ] This gives another angle from which to tackle the problem.

Solution 1

Solution 2

Thinking Back

     Both solutions are related in the sense that they hinge on some form of the relation maked as [*].  That led to an equation in  a.  Once a  is found,  k  can be found, and the solutions proceed similarly.  The equation [*] is a manifestation of the fact that for a repeated root, both P(x) and P’(x) share a common factor.