Difference between revisions of "2015 UNCO Math Contest II Problems/Problem 3"
(Created page with "== Problem == If P is a polynomial that satisfies <math>P(x^2 +1) = 5x^4 +7x^2 +19</math>, then what is <math>P(x)</math>? (Hint: <math>P</math> is quadratic.) == Solution == ...") |
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== Solution == | == Solution == | ||
| + | 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</math>. 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}</math> | ||
== See also == | == See also == | ||
Latest revision as of 09:03, 6 March 2024
Problem
If P is a polynomial that satisfies 
, then what is 
? (Hint: 
is quadratic.)
Solution
Define 
, so 
. By matching coefficients (the coefficients of each power of x on both sides must be equal), we derive the system 
,
,and 
, from which we see 
and 
. Thus, 
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 | ||
