Nowhere as hard as it looks -- See also Vieta's formulas.
What is the largest root of this equation?
x 8 − 1 2 4 x 7 + 6 4 9 6 x 6 − 1 8 7 4 0 4 x 5 + 3 2 4 8 1 5 4 x 4 − 3 4 5 2 8 1 1 6 x 3 + 2 1 9 0 2 5 0 6 4 x 2 − 7 5 4 9 1 1 2 3 6 x + 1 0 7 8 2 8 2 2 0 5 = 0
The roots are integers.
Hint: See Vieta's formulas .
The answer is 29.
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1 solution
We do not know that the 8 prime factors of 1 0 7 8 2 8 2 2 0 5 are, in fact, the 8 roots of the polynomial. It is entirely possible that some of the roots are 1 . Or that the roots are not integers.
Vieta's formulas simply tell us that the 8 roots of the equation multiply to 1 0 7 8 2 8 2 2 0 5 . But maybe 7 of them are 1 and the eighth is 1 0 7 8 2 8 2 2 0 5 . Maybe some are fractions. Maybe some are irrational or complex.
For example, 1 0 has two prime factors, but x 2 − 1 1 x + 1 0 = ( x − 1 ) ( x − 1 0 ) does not have 2 and 5 as roots.
As another example, your proposed solution would apply to any degree 8 polynomial with constant term 1 0 7 8 2 8 2 2 0 5 . So, for example, x 8 − 1 0 7 8 2 8 2 2 0 5 = 0 . But that polynomial has irrational roots.
In order to use Vieta's formulas to find the roots, we need to either use all coefficients (a system of 8 equations and 8 variables that is virtually impossible to solve without a computer) or know some additional information about the properties of the roots.
And, yes, I do have a computer algebra system on my computer and on my cell phone (Maxima on Android).
If we are given that all the roots are integers, and the factorisation of 1078282205, the problem is simple without a computer:
- One can see that the last term is greater than the absolute value of all the others combined, hence − 1 and 1 are not roots.
Therefore the absolute values of the roots are the 8 prime numbers in the factorisation: 5 , 7 , 1 1 , 1 3 , 1 7 , 1 9 , 2 3 , 2 9 .
- Then we can see that the greatest sum of the roots is 1 2 4 when all roots are positive. Hence the roots are 5 , 7 , 1 1 , 1 3 , 1 7 , 1 9 , 2 3 , 2 9 . So 2 9 is the greatest.
The degree of polynomial is 8. Therefore, it has 8 roots. The constant term has 8 prime factors. Therefore, ignoring their signs, the factors are the roots.
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You need to know that 1 or − 1 is not a root first. ( x − 1 ) ( x − 3 5 ) ( x − 1 1 ) ( x − 1 3 ) ( x − 1 7 ) ( x − 1 9 ) ( x − 2 3 ) ( x − 2 9 ) has the same constants term etc.
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Try factoring the polynomial. I added the statement that the roots were integers.
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@A Former Brilliant Member – Yes, I agree with the answer, but you can't go from knowing the constant term factorizes into 8 primes straight to those primes being the roots (up to choice of sign), without eliminating 1 and − 1 as roots.
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@Alex Burgess – I like the statement that the roots are integers in the problem.
The 1 and − 1 shouldn't be added to the question of cause, just as a justification in a solution.
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@Alex Burgess – N. B., In the solution, I said, "assuming that the prime factors were the solution (I agree that they might not be). That was followed by a direct verification of the solution.
If you factor the constant term you will discover that it is the product of 8 primes: 3, 5, 7,11, 13, 17, 19, 23 and 29. From Vieta's formulas, you know that those are the roots of the polynomial. The largest root is 29. That is the answer.
Using Vieta's and assuming that the prime factors were the solution (I agree that they might not be):
1 2 3 4 5 6 7 8 1 2 4 6 4 9 6 1 8 7 4 0 4 3 2 4 8 1 5 4 3 4 5 2 8 1 1 6 2 1 9 0 2 5 0 6 4 7 5 4 9 1 1 2 3 6 1 0 7 8 2 8 2 2 0 5
The coefficients match. Problem solved.
The problem can be done easily on a smartphone. Maxima on Android is free. It can solve the problem. I just did it. Also, Wolfram/Alpha can do it; but, Wolfram/Alpha is not free.