Question 486603: If x^2+x-1=0, what is x^3 + 2x^2 + 2011 closest to?
choose the answer and explain the reason to choose your response
2011
2010
2013
2012
Found 2 solutions by richard1234, chessace:
Answer by richard1234(7193)   (Show Source):
Answer by chessace(471)  (Show Source):
You can put this solution on YOUR website! This problem is ambiguous, but both interpretations lead to the same result, so it's not that bad. It also shows why "explain reason" is meaningful; there are 2 valid reasons for the same answer.
Interpretation 1:
The wording makes you think there is an implied "roots of" (...closest to...).
In that case, solve for x in given quadratic, x = .5(-1+-SQRT(5)).
Note that x^3 + 2x^2 + 2011 = x^2(x+2) + 2011.
Then plug the 2 solutions above:
x^2 = .5(3 -+ sqrt(5)) [multply it out very carefully]
The -+ means that when the + root is chosen, this term is -, and vice versa.
x+2 = .5(3 +- sqrt(5)) [not as much effort]
So their product = .25(9 +-0 -5) = 1
This give a (triple) root at 2012, the final answer.
Interpretation 2:
Compare f(x) = x^3 + 2x^2 + 2011 to the various constant functions
g(x) = 2011 etc.
We want to "factor" f(x) using (x^2 + x -1), so rather than guessing, do the long division f(x) / (x^2 + x -1) to get quotient (x + 1) with remainder 2012 (coincidence?).
Now consider h(x) = (x + 1)(x^2 + x -1)
Expand: h(x) = x^3 + 2x^2 -1
Thus f(x) = h(x) + 2012
But also h(x) = (x + 1) * 0 due to it's 2nd factor = 0 by given equation.
So f(x) = 2012 as a constant function.
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