Difference between revisions of "2023 SSMO Relay Round 2 Problems/Problem 1"
(Created page with "==Problem== Consider the cubic polynomial <math>P(x)=ax^3+bx^2+cx+d</math>, where <math>a,b,c,d</math> are single-digit integers, which has roots of approximately <cmath>x \ap...") |
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==Solution== | ==Solution== | ||
| + | Note that the sum of the three roots is approximately <math>0.71\approx\frac{5}{7}</math>. So, we have <math>a=7</math>. Therefore <cmath>|f(3)| \approx 7(3-(-0.95))(3-0.2)(3-1.46)\approx\boxed{119}.</cmath> | ||
| + | |||
| + | ~pinkpig | ||
Latest revision as of 15:30, 15 September 2025
Problem
Consider the cubic polynomial 
, where 
are single-digit integers, which has roots of approximately
![\[x \approx -0.9518399, 0.2055095, 1.460616.\]](http://latex.artofproblemsolving.com/1/3/0/130e48e7d7e9de3138391c20ebecd1488e8319f2.png)
Compute 
.
Solution
Note that the sum of the three roots is approximately 
. So, we have 
. Therefore ![\[|f(3)| \approx 7(3-(-0.95))(3-0.2)(3-1.46)\approx\boxed{119}.\]](http://latex.artofproblemsolving.com/7/e/8/7e8d03ada127abb9cd09e99f437a90f23b61491d.png)
~pinkpig
