Difference between revisions of "2007 Alabama ARML TST Problems/Problem 14"
(New page: ==Problem== Find the sum of the ''real'' roots of <cmath>x^6+145x^4-2007x^3+145x^2+1=0.</cmath> ==Solution== === Step 1: using the symmetry === Let <math>p(x)=x^6+145x^4-2007x^3+145x^2...) |
(→Step 5: Handling the remaining part of p: fixed a small mistake) |
||
Line 60: | Line 60: | ||
To prove this, one can for example write: | To prove this, one can for example write: | ||
− | <cmath>r(x) = (x^4 + 9x^3 + | + | <cmath>r(x) = \left(x^4 + 9x^3 + \frac{81}{4}\cdot x^2\right) + \left(\frac{81}{4}\cdot x^2 + 9x + 1\right) + \left(225-2\cdot\frac{81}{4}\right) x^2 </cmath> |
+ | |||
+ | <cmath>r(x) = x^2\left(x+\frac92\right)^2 + \left(\frac{9x}2+1\right)^2 + \frac{369}2\cdot x^2</cmath> | ||
=== Step 6: Summary === | === Step 6: Summary === |
Latest revision as of 07:09, 26 January 2009
Contents
Problem
Find the sum of the real roots of
Solution
Step 1: using the symmetry
Let
The polynomial is symmetric. Each symmetric polynomial has the following property: for , is a root if and only if is.
To see why this is true, assume that we have a such that . We can now divide both sides by . We get:
But the left hand side is just . Thus also .
Step 2: multiplying three quadratic terms
Our polynomial has six complex roots: , , , , , and . It can be expressed as:
Let , , and . We can then write:
We can now compare the coefficients of , and in this expression. (Comparing other coefficients is not necessary due to symmetry.) We get the following equations:
Simplifying, we get:
Using Vieta's formulas, this means that , and are the roots of the polynomial .
Step 3: Finding a root of q
Obviously, is increasing for , and . Thus has exactly one positive root. We can easily find that . Thus WLOG .
(Note: Remembering that and , we now have and . From the first equation, let . Substituting into the second one, we get . This obviously has no real solution, thus and are not real. But we don't need this observation.)
Step 4: Partially dividing p
We now know that is a factor of . We can partially divide to get:
Step 5: Handling the remaining part of p
The polynomial has no real roots.
To prove this, one can for example write:
Step 6: Summary
We found that , where has no real roots. Thus all real roots of are the two real roots of , and their sum is obviously .
Note
roots of p computed by Octave: -4.48006 + 14.30270i -4.48006 - 14.30270i 8.88748 + 0.00000i 0.11252 + 0.00000i -0.01994 + 0.06367i -0.01994 - 0.06367i
See also
2007 Alabama ARML TST (Problems) | ||
Preceded by: Problem 13 |
Followed by: Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |