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Math Correction

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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 -

    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 -

    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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