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 |