2009 UNCO Math Contest II Problems/Problem 7
Problem
A polynomial 
has a remainder of 
when divided by 
and a remainder of 
when divided
by 
. What is the remainder when is divided by 
?
Solution
Since we're being asked to find a remainder when a polynomial is divided by a quadratic, we can assume that the remainder will be at most linear. Thus, the remainder can be written in the form 
.
It is given that the polynomial has a remainder of when divided by and a remainder of when divided by , which translates to 
and 
. However, for both of these equations to always be true, the coefficient 
must be equal to 
.
Thus, 
and 
.
These equations simplify to 
and 
, which shows that
, so
,
meaning 
.
Plugging back into either equation gives 
, meaning the remainder is 
.
See also
| 2009 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
