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
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
It gives practice in manipulation of polynomials.
Possible approach
An easy lesson starter!
Key question
What is $p(x)-q(x)$?
Possible extension
Learners can make up their own problems by writing down two polynomials in $x$ and then eliminating $x$ between the expressions. They might be asked to make up such a problem and exchange problems with their partner. Then they can compare and check results in pairs.
Possible support
Try a simpler example such as: find the formula relating
$p$ and $q$ where $p=x+3$ and $q=x^2$.