Difference between revisions of "2009 UNCO Math Contest II Problems/Problem 7"
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== Problem == | == Problem == | ||
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A polynomial <math>P(x)</math> has a remainder of <math>4</math> when divided by <math>x+2</math> and a remainder of <math>14</math> when divided | A polynomial <math>P(x)</math> has a remainder of <math>4</math> when divided by <math>x+2</math> and a remainder of <math>14</math> when divided | ||
| − | by <math>x-3 | + | by <math>x-3</math>. What is the remainder when <math>P(x)</math> is divided by <math>(x+2)(x-3)</math>? |
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== Solution == | == Solution == | ||
Latest revision as of 12:25, 23 December 2019
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 | ||
