JOMO 7, Short 10

Algebra Level 4

$\large \displaystyle x^{x^5-37x^4+7x^3+84x^2+2x+9}=1.$

Find the sum of all distinct values of $x$ (including complex numbers) that satisfy the equation above.

The answer is 37.

3 solutions

Jordi Bosch
Sep 14, 2014

For simplicity consider the equation $x^{a} = 1$ We can have $x = 1, x = -1$ with $a = 2n, a = 0$ . These 3 cases are all the possibilities. We aply it to the given equation. We see when $x = -1$ the exponent is even, so $x = -1$ is posible. Notice we are not given any restriction over $x$ , so we know the exponent will have 5 complex roots with sum equal to $-\frac{-37}{1} = 37$ Therefore the desired sum is $1 - 1 + 37 = \boxed{37}$

Note, you should also check that the roots of the quintic are distinct, and not equal to 1 or -1.

Calvin Lin Staff - 6 years, 6 months ago

nice way , excatly same !!

math man - 6 years, 8 months ago

Department 8
Jul 12, 2015

We see that equation given would be satisfied with any value of $x$ in the exponent equation is $x$ is equal to the he roots of equation therefore the sum of roots in given equation will be 37

Ashu Dablo
Sep 27, 2014

use vieta's

JOMO 7, Short 10

The answer is 37.

3 solutions

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