Posted by **Amanda** on Wednesday, December 8, 2010 at 7:36pm.

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**, Wednesday, December 8, 2010 at 7:59pm
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 non-negative over its domain, ℝ.

The factors of the denominator have similar properties, non-negative throughout its domain, ℝ.

Since the function is a non-negative number divided by a non-negative number, there are no vertical asymptotes, nor does it cross the x-axis.

- Correction -
**MathMate**, Wednesday, December 8, 2010 at 8:35pm
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.

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