2015 UNCO Math Contest II Problems/Problem 3


If P is a polynomial that satisfies $P(x^2 +1) = 5x^4 +7x^2 +19$, then what is $P(x)$? (Hint: $P$ is quadratic.)


Define $P(x)=ax^2+bx+c$, so $a(x^2+1)^2+b(x^2+1)+c=ax^4+2ax^2+a+bx^2+b+c=5x^4+7x^2+19$. By matching coefficients (the coefficients of each power of x on both sides must be equal), we derive the system $a=5$,$2a+b=7$,and $a+b+c=19$, from which we see $b=-3$ and $c=17$. Thus, $P(x)=\boxed{5x^2-3x+17}$

See also

2015 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
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All UNCO Math Contest Problems and Solutions