Strange curves!?

Question

take the formula (X^3+n^3)/(x+n). Take values of x (for arguments sake) from-20 to 20, and for each value of x plot a curve for values of n from -20 to 20. In each case you'll get a lovely smooth curve for every point except where x = -n. At this point you will be dividing by zero so the curve will spike to infintiy. At each point where the spike happens you can predict what the output value of the formula should be to keep the curve smooth. For example where x=1 and n=-1, an output value of 3 would produce a perfect smooth curve. Similarly where x=2 and n=-2, a value of 12 would keep the curve smooth. Am I missing something? Have I made a mistake in the way I'm calculating? Or is this just genuinely a weird set of curves?

Answer 1

The differences here all come down to the fact that ypiu can factor an (x+n) term out of (x^3 + n^3) to get (x^2 - xn + n^2)(x+n)/(x+n), and this can be reduced to x^2 - xn + n^2.

Now keep in mind this doesn't mean the function IS also x^2 - xn + n^2. You have to keep those discontinuities in mind. The functin is still really (x^3 + n^3) / (x+n). However, the function BEHAVES like x^2 - xn + n^2 except for the point x = -n.

Answer 2

In this case the curve won't spike at x = -n. There will simply be a "hole" in the domain there.

(x³ + n³) / (x + n)

= [(x + n)(x² - xn + n²)] / (x + n)

= x² - xn + n²

This is the equation of a smooth curve, a parabola, with the single point x = -n removed from the domain. If you will, it is a "hole" in the curve. You will get a similar hole in the domain any time a factor in the denomintor is cancelled by the same term in the numerator, provided that there is a value for x at which the factor equals zero.

Answer 3

You have to factorize the numerator then cancel :

( x + n )

You'll have only polynomial term ,

Now plot the curve .