2015 UNCO Math Contest II Problems/Problem 3

Problem

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.)

Solution

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, $P(x)=\boxed{5x^2-3x+17}$

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
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2015 UNCO Math Contest II Problems/Problem 3

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