Polynomial Relations
Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.
Problem
Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials.
Let $p(x) = x^2 + 2x$ and $q(x) = x^2 + x + 1$. Then, using a method which does not depend on knowing the answer, show that the relationship between the polynomials is:
\[ p^2 - 2pq + q^2 + 3p - 4q + 3 = 0 \]
Getting Started
What is $p-q$?
Student Solutions
Good solutions to this problem were received from Tyrone of Cyfarthfa High School in Merthyr Tydfil, and Koopa of Boston College in the USA.
Tyrone solved the problem by relating both polynomials to $(x+1)^2$ :
But $x=(p+1)^{1/2}-1$ (from eqn (1)). So
$$ \eqalign { \Rightarrow p+1&=&q+ ((p+1)^{1/2}-1) \\
&=&q+ (p+1)^{1/2}-1 }$$
$$ \eqalign { p-q+2&=&(p+1)^{1/2} \\
(p-q+2)^2&=&p+1}$$.
Squaring the bracket,
$$ \eqalign { &p^2-pq+2p-pq+q^2-2q+2p-2q+4=p+1 \\ &p^2 -2pq+q^2+4p-4q+4=p+1 \\ &p^2-2pq+q^2+3p-4q+3=0 } $$.
Teachers' Resources
Why do this problem?
It gives practice in manipulation of polynomials.
Possible approach
An easy lesson starter!
Key question
What is $p(x)-q(x)$?
Try a simpler example such as: find the formula relating
$p$ and $q$ where $p=x+3$ and $q=x^2$.