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
All UNCO Math Contest Problems and Solutions

Page

Toolbox

Search

2015 UNCO Math Contest II Problems/Problem 3

Problem

Solution

See also