Analyze the function: x^2 + 3 / X + 1
X Intercepts:
Y Intercepts:
Asymptotes:

Descartes Rule: f(x) = -X^4 + 3X^2 + X + 7
How do I find the positive and negative zeros?

1.



Put x = 0 to find the y intercepts. (y = 3)
Put y = 0 to find the x intercepts. (x = no real solution)

At the asymptote, you'll have the denominator either as infinity, negative infinity or zero.

This way, you get the line y = 1 as an asymptote (when x tends to infinity, y tends to 1), the line y = -1 as an asymptote (when x tends to negative infinity, y tends to -1) and when x = -1, you get a zero denominator, making this time a vertical asymptote at x = -1.

Sorry if I made a mistake in my calculations, it's really late here... shouldn't be here that late...

Descartes rule of signs says that the number of positive real roots is either equal to the number of changes in sign of f(x) or less than that by an even number.

How many times does it change sign? 1 time.

So, there must only be 1 positive real root because we can not be less than 1 by an even integer.

The rule also says that the number of negative real roots is equal to the number of changes in sign of f(-x) or less than that by an even number.

How many sign changes are in f(-x)? Sub in -x for x and we get



How many sign changes now? 3.

So, there are either 3 or 1 negative roots. See?

Solving, f(x) has 1 positive real root and 1 negative real root, as DRS said. The other 2 are complex.