#5**+5 **

Thanks Geno,

I am just going to work through my version of your logic. :)

$$\\x^{15}+x^5+2=0\\\\

Let\;\;y=x^5\\\\

y^3+y+2=0\\\\

y(y^2+1)=-2\\\\

y^2+1 $ must be positive so $y$ must be negative. Hence$\\\\

x^5$ must be negative, hence $\\\\

x $ must be negative$\\\\

$Therefore, no positive roots exist.$$$

Melody
Sep 29, 2014

#1**0 **# prove that x^15+x^5+2=0, do not have positive roots?

Let's write x^5 = y

Then we get:

y^3 +y +2 = 0

y^3 +y = -2

y(1 +y^2) = -2

y =-2 or y^2 +1 =-2

We solve y^2 +1 =-2

y^2 = -3

y doesn't have any solution among the real numbers R.

y = √-3 = √3 *i or -√3 *i

To sum it up: y = -2 or y = √3 *i or y = -√3 *i

x^5 = y

...

Guest Sep 28, 2014

#2**0 **

Let's write x^5 = y good idea

Then we get:

y^3 +y +2 = 0

y^3 +y = -2

y(1 +y^2) = -2 Yes

y =-2 or y^2 +1 =-2 WHY?

Melody
Sep 28, 2014

#4**+5 **

I would argue that if you want the equation to have a positive root, then x must be positive.

But if x is positive, then both x^15 is positive and x^5 is positive.

Adding two positive number to the number 2, your answer must be positive; thus, it can't be zero.

(proof by contradiction)

geno3141
Sep 28, 2014

#5**+5 **

Best Answer

Thanks Geno,

I am just going to work through my version of your logic. :)

$$\\x^{15}+x^5+2=0\\\\

Let\;\;y=x^5\\\\

y^3+y+2=0\\\\

y(y^2+1)=-2\\\\

y^2+1 $ must be positive so $y$ must be negative. Hence$\\\\

x^5$ must be negative, hence $\\\\

x $ must be negative$\\\\

$Therefore, no positive roots exist.$$$

Melody
Sep 29, 2014