+0

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

0
556
5

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

Sep 28, 2014

#5
+95360
+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.\$\$\$

.
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

...

Sep 28, 2014
#2
+95360
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?

Sep 28, 2014
#3
+5

You are right, Melody!

I must have been asleep :P

Ok, I'll try once more:

Let's write x^5 = y

Then we get:

y^3 +y +2 = 0

y^3 +y = -2

y(1 +y^2) = -2

Ok, we know that:

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

I'm not really sure from here :(

Sep 28, 2014
#4
+17747
+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.

Sep 28, 2014
#5
+95360
+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