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2007 Alabama ARML TST Problems/Problem 6

Problem

If $r$ is a root of $x^2+x+6$, then compute the value of

\[r^3+2r^2+7r+17.\]

Solution

$r$ satisfies $r^2+r+6=0$. Thus, $r^3+2r^2+7r+17 = (r^3+r^2+6r)+(r^2+r+6)+11 =$ $(r+1)(r^2+r+6)+11 = (r+1)(0)+11 = 11$.

See also

2007 Alabama ARML TST (Problems)
Preceded by:
Problem 5 		Followed by:
Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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