Hexotic Equation - 2

Algebra Level 4

Find the sum of all rational roots of the equation 6 x 6 − 25 x 5 + 31 x 4 − 31 x 2 + 25 x − 6 = 0 6x^6-25x^5+31x^4-31x^2+25x-6=0


The answer is 2.5.

View wiki
Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Chew-Seong Cheong
Jul 24, 2015

f ( x ) = 6 x 6 − 25 x 5 + 31 x 4 − 31 x 2 + 25 x − 6 = 6 ( x 6 − 1 ) − 25 x ( x 4 − 1 ) + 31 x 2 ( x 2 − 1 ) = 6 ( x 2 − 1 ) ( x 4 + x 2 + 1 ) − 25 x ( x 2 − 1 ) ( x 2 + 1 ) + 31 x 2 ( x 2 − 1 ) = ( x 2 − 1 ) ( 6 x 4 − 25 x 3 + 37 x 2 − 25 x + 6 ) = ( x 2 − 1 ) ( 6 x 4 − 15 x 3 + 6 x 2 − 10 x 3 + 25 x 2 − 10 x + 6 x 2 − 15 x + 6 ) = ( x 2 − 1 ) ( 3 x 2 ( 2 x 2 − 5 x + 2 ) − 5 x ( 2 x 2 − 5 x + 2 ) + 3 ( 2 x 2 − 5 x + 2 ) ) = ( x + 1 ) ( x − 1 ) ( 2 x − 1 ) ( x − 2 ) ( 3 x 2 − 5 x + 3 ) \begin{aligned} f(x) & = 6x^6-25x^5+31x^4 - 31x^2 + 25x - 6 \\ & = 6(x^6 - 1) -25x(x^4-1) +31x^2(x^2-1) \\ & = 6(x^2 - 1)(x^4+x^2+1) -25x ( x^2 -1)(x^2+1) +31x^2(x^2-1) \\ & = (x^2-1) (6x^4 -25x^3 + 37x^2 - 25x +6) \\ & = (x^2-1) (6x^4 -15x^3 + 6x^2 -10x^3+ 25x^2-10x + 6x^2 -15x +6) \\ & = (x^2-1) (3x^2(2x^2 -5x + 2) - 5x(2x^2 -5x + 2)+ 3(2x^2 -5x + 2)) \\ & = (x+1)(x-1)(2x-1)(x-2)(3x^2-5x+3) \end{aligned}

Therefore the real rational roots of f ( x ) f(x) are − 1 -1 , 1 2 \frac{1}{2} , 1 1 and 2 2 and their sum is − 1 + 1 2 + 1 + 2 = 2.5 -1+\frac{1}{2}+1+2 = \boxed{2.5} .

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Aakash Khandelwal
Jul 24, 2015

More better approach can be by dividing by x^3 Short and simple!!!

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Can you add more details? How does that make it short and simple?

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...