2018 AMC 12A Problems/Problem 21
Contents
Problem
Which of the following polynomials has the greatest real root?
Solution 1
We can see that our real solution has to lie in the open interval 
. From there, note that 
if 
, 
are odd positive integers if 
, so hence it can only either be B or E(as all of the other polynomials will be larger than the polynomial B). Finally, we can see that plugging in the root of 
into B gives a negative, and so the answer is 
. (cpma213)
Solution 2 (Calculus version of solution 1)
Note that 
and 
. Calculating the definite integral for each function on the interval ![$[-1,0]$](http://latex.artofproblemsolving.com/3/1/d/31d4af7eb18014c3073c9fa861cdbfc4afb367a0.png)
, we see that 
gives the most negative value. To maximize our real root, we want to maximize the area of the curve under the x-axis, which means we want our integral to be as negative as possible and thus the answer is .
Solution 3 (Alternate Calculus Version)
Newton's Method is used to approximate the zero 
of any real valued function given an estimation for the root 
: 
After looking at all the options, 
gives a reasonable estimate. For options A to D, 
and the estimation becomes 
Thus we need to minimize the derivative, giving us B. Now after comparing B and E through Newton's method, we see that B has the higher root, so the answer is . (Qcumber)
See Also
| 2018 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 20 |
Followed by Problem 22 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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