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1971 Canadian MO Problems/Problem 4: Difference between revisions

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== Problem ==
== Problem ==
Determine all real numbers <math>\displaystyle a</math> such that the two polynomials <math>\displaystyle x^2+ax+1</math> and <math>\displaystyle x^2+x+a</math> have at least one root in common.
Determine all real numbers <math>a</math> such that the two polynomials <math>x^2+ax+1</math> and <math>x^2+x+a</math> have at least one root in common.


== Solution ==
== Solution ==


Let this root be <math>\displaystyle r</math>.  Then we have
Let this root be <math>r</math>.  Then we have


<center>
<center>
<math>\displaystyle \begin{matrix} r^2 + ar + 1 &=& r^2 + r + a\\
<math>\begin{matrix} r^2 + ar + 1 &=& r^2 + r + a\\
ar + 1 &=& r + a\\
ar + 1 &=& r + a\\
(a-1)r &=& (a-1)\end{matrix} </math>
(a-1)r &=& (a-1)\end{matrix} </math>
</center>
</center>


Now, if <math>\displaystyle a = 1 </math>, then we're done, since this satisfies the problem's conditions.  If <math>\displaystyle a \neq 1</math>, then we can divide both sides by <math>\displaystyle (a - 1) </math> to obtain <math>\displaystyle r = 1 </math>.  Substituting this value into the first polynomial gives
Now, if <math>a = 1 </math>, then we're done, since this satisfies the problem's conditions.  If <math>a \neq 1</math>, then we can divide both sides by <math>(a - 1) </math> to obtain <math>r = 1 </math>.  Substituting this value into the first polynomial gives


<center>
<center>
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It is easy to see that this value works for the second polynomial as well.
It is easy to see that this value works for the second polynomial as well.


Therefore the only possible values of <math>\displaystyle a </math> are <math>\displaystyle 1 </math> and <math>\displaystyle -2 </math>.  Q.E.D.
Therefore the only possible values of <math>a </math> are <math>1 </math> and <math>-2 </math>.  Q.E.D.
 
----
* [[1971 Canadian MO Problems/Problem 3|Previous Problem]]
* [[1971 Canadian MO Problems/Problem 5|Next Problem]]
* [[1971 Canadian MO Problems|Back to Exam]]


{{Old CanadaMO box|num-b=3|num-a=5|year=1971}}
[[Category:Intermediate Algebra Problems]]
[[Category:Intermediate Algebra Problems]]

Revision as of 21:48, 17 November 2007

Problem

Determine all real numbers $a$ such that the two polynomials $x^2+ax+1$ and $x^2+x+a$ have at least one root in common.

Solution

Let this root be $r$. Then we have

$\begin{matrix} r^2 + ar + 1 &=& r^2 + r + a\\ ar + 1 &=& r + a\\ (a-1)r &=& (a-1)\end{matrix}$

Now, if $a = 1$, then we're done, since this satisfies the problem's conditions. If $a \neq 1$, then we can divide both sides by $(a - 1)$ to obtain $r = 1$. Substituting this value into the first polynomial gives

$\begin{matrix} 1 + a + 1 &=& 0\\ a &=& -2 \end{matrix}$

It is easy to see that this value works for the second polynomial as well.

Therefore the only possible values of $a$ are $1$ and $-2$. Q.E.D.

1971 Canadian MO (Problems)
Preceded by
Problem 3 		1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • Followed by
Problem 5
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