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Problem of the Week

Problem 34

This is our last question before the summer holidays start. It is designed to be long and difficult, so feel free to submit this in chunks – we are happy to provide hints throughout.

These questions are all designed around polynomials.

Question 1 – The Common Chord

Two quadratic graphs intersect at two distinct points.
The line joining these two points is called their common chord.

(a) The graphs

P(x)=x2+2x+1andQ(x)=3x2−4x+5P(x)=x^2+2x+1 \\ and \\Q(x)=3x^2-4x+5

intersect at two points. Find the equation of the line joining the two intersection points.

(b) The graphs

P(x)=2x2+3x−1andQ(x)=5x2−4x+8P(x)=2x^2+3x-1\\ and \\Q(x)=5x^2-4x+8

intersect at two points. The x-coordinates of the intersections are r and s. Without solving the quadratic equation, find the gradient of the line joining the two points.

(c) Let

P(x)=ax2+bx+candQ(x)=dx2+ex+fP(x)=ax^2+bx+c \\and\\ Q(x) = dx^2 + ex + f

where a\neq d.

The graphs intersect at points whose x-coordinates are r and s.

Show that r and s satisfy

(a−d)x2+(b−e)x+(c−f)=0.(a-d)x^2+(b-e)x+(c-f)=0.

(d) The quadratic from part (c) can be written as (a-d)(x-r)(x-s)=0.

Expand this expression and compare coefficients to show that

r+s=e−ba−d. r+s=\frac{e-b}{a-d}.

(e) The gradient of the common chord is

m=P(s)−P(r)s−r.m=\frac{P(s)-P(r)}{s-r}.

Show that m=a(r+s)+b.

Hence prove that

m=ae−bda−d.m=\frac{ae-bd}{a-d}.

(f) The common chord has equation y=mx+k. Show that

k=af−cda−d.k=\frac{af-cd}{a-d}.

Hence prove that the equation of the common chord is

y=(ae−bd)x+(af−cd)a−dy=\frac{(ae-bd)x+(af-cd)}{a-d}

(g) Two quadratic graphs have common chord y=4x+7.

One graph is

y=3x2−2x+5.y=3x^2-2x+5.

Find one possible equation for the second quadratic.

Question 2 – The Hidden Derivative (Further Maths Recommended)

When (x) changes to (x+h), expressions involving (x) also change.

In this investigation, you will explore the coefficient of h in various expansions.

(a) Expand each expression and state the coefficient of h^1.

(x+h)2and(x+h)3(x+h)^2\\ and \\(x+h)^3

Hence predict the coefficient of h^1 in

(x+h)8.(x+h)^8.

(b) A polynomial (P(x)) has the property that the coefficient of h^1 in P(x+h) is

12x3−8x.12x^3-8x.

Find one possible polynomial (P(x)).

(c) Without fully expanding, find the coefficient of h^1 in

    \[ 3(x+h)^9 -2(x+h)^6 +5(x+h)^4 -(x+h)^2 +17. \]

Question 3 – Consecutive Squares

Throughout this question, n denotes a positive integer.

(a) By expanding (n+1)^2 or otherwise, show that n^2+n is never a perfect square.

(b) Show that

(x2+1)2=x4+2x2+1 (x^2+1)^2=x^4+2x^2+1

and

(x2+x)2=x4+2x3+x2.(x^2+x)^2=x^4+2x^3+x^2.

(c*) Let

N=n4+n3+n2+n+1.N=n^4+n^3+n^2+n+1.

By comparing N with suitable consecutive perfect squares, determine all positive integers n for which N is itself a perfect square.

You may find it useful to split into odd and even cases.

You should justify why no other values of n are possible.

Happy Holidays!

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