Difference between revisions of "2015 UNCO Math Contest II Problems/Problem 3"
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== Solution == | == Solution == | ||
| − | <math>P(x)=5x^2- | + | Define <math>P(x)=ax^2+bx+c</math>, so <math>a(x^2+1)^2+b(x^2+1)+c=ax^4+2ax^2+a+bx^2+b+c=5x^4+7x^2+19. By matching coefficients (the coefficients of each power of x on both sides must be equal), we derive the system </math>a=5<math>,</math>2a+b=7<math>,and </math>a+b+c=19<math>, from which we see </math>b=-3<math> and </math>c=17<math>. Thus, </math>P(x)=\boxed{5x^2-3x+17}$ |
== See also == | == See also == | ||
Revision as of 10:02, 6 March 2024
Problem
If P is a polynomial that satisfies 
, then what is 
? (Hint: 
is quadratic.)
Solution
Define 
, so 
a=5
2a+b=7
a+b+c=19
b=-3
c=17
P(x)=\boxed{5x^2-3x+17}$
See also
| 2015 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
