Posted by Amanda on .
Explain how you can tell(without graphing) that the function has no x intercept and no asymptotes. What is the end behaviour?
r(x)= x^6+10 / x^4+8x^2+15
How would I solve this? I know it has something to do with factoring but im not sure what else to do.
this is what i have so far:
r(x)= x^6+10 / (x^2+3)(x^2+5)
Can someone help me finish? and explain how the answer? thanks

Math Correction 
MathMate,
Good start!
We will examine the numerator, x^6+10.
It is a monotonically increasing function, the minimum value of which is 10 when x=0. So it is nonnegative over its domain, ℝ.
The factors of the denominator have similar properties, nonnegative throughout its domain, ℝ.
Since the function is a nonnegative number divided by a nonnegative number, there are no vertical asymptotes, nor does it cross the xaxis. 
Correction 
MathMate,
Sorry, there is a correction:
"It is a monotonically increasing function..."
should read
"It is an even function..."
The arguments and conclusions do not change.