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 |