Solve without a calculator – 7

Algebra Level 2

Find the absolute value of the product of all (complex) roots of the polynomial

f ( x ) = 1 7 x 6 − 534 7 x 4 + 11487 x 2 − 396508 f(x) = \frac 17 x^6 - \frac {534}7 x^4 + 11487 x^2 -396508


The answer is 2775556.

Henry U by Henry U , 17, Germany

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2 solutions

Parth Sankhe
Nov 17, 2018

Product of roots = constant term ÷ x n x^n coefficient

Therefore, product = 396508×7

2 Helpful 0 Interesting 0 Brilliant 0 Confused

I have edited the problem so that +2775556 is now correct. Thanks for noting the mistake and I'm sorry for the confusion.

Henry U - 2 years, 6 months ago
Levi Walker
Nov 25, 2018

All roots satisfy 1 7 x 6 − 534 7 x 4 + 11487 x 2 − 396508 = 0 \frac{1}{7}x^6 - \frac{534}{7}x^4 + 11487x^2 - 396508 = 0 , or equivalently x 6 − 534 x 4 + 80409 x 2 − 2775556 = 0 x^6 - 534x^4 + 80409x^2 - 2775556 = 0 . The factorization of any generic polynomial is ( x − r 1 ) ( x − r 2 ) ⋯ ( x − r n ) = x n + ⋯ + ( − 1 ) n ∏ i = 1 n r i (x-r_1)(x-r_2)\cdots (x-r_n) = x^n + \cdots + (-1)^n \prod_{i=1}^n r_i

Thus, the product of all roots is the constant term, or − 2775556 -2775556 .

(I'm not sure why complex roots were specified, though, since all of the roots of this polynomial turn out to be real.)

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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