Question

find the domain, x and y-intercepts, vertical and horizantal asymptotes for the rational function

y=(x^3-27)/x^4-2x^3+9x^2-18x

Answer 1

@jim_thompson5910 Could you help me with this?

Answer 2

\[\Large y=\frac{x^3-27}{x^4-2x^3+9x^2-18x}\]
right?

Answer 3

@Mhm120: jim_thompson asked you a question. If you confirm the expression he has written is correct he can continue with the answer.

Answer 4

Correct

Answer 5

and sorry was the late reply

Answer 6

Factor the numerator and the denominator.
For the numerator use the identity: \(a^3 - b^3 = (a-b)(a^2 - ab + b^2)\).
For the denominator factor out x. Try a few small values of x such as x = 1, -1, 2, -2, etc to see if you can find a root of the remaining expression to see if you can find another factor.

Answer 7

\(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\) (plus ab not minus ab)

Answer 8

sorry, but i don't get the part where you're using the small values to find the remaining expression

Answer 9

\( x^4 - 2x^3 + 9x^2 - 18x = x(x^3 - 2x^2+9x-18) \)
We want to see if \((x^3 - 2x^2+9x-18) \) can be factored. Try to find if x = -1, +1, -2, +2 will make the expression zero. If x = a makes it zero then (x-a) is a factor.

Answer 10

Once we reduce it to a quadratic we have factorization or quadratic formula that we can use to find further real factors, if any. Using the identity we already have a quadratic expression in the numerator.

Answer 11

oh okay thanks!!

Answer 12

You are welcome.

\( \Large
y=\frac{x^3-27}{x^4-2x^3+9x^2-18x} = \frac{(x-3)(x^2+3x+9)}{x(x-2)(x^2+9)}
\)

There are no other real factors but there are complex factors which we don't have to worry about.

Now you can easily find the domain, x-intercepts, y-intercepts, etc.

Answer 13

oh okay got it thankyou so much

Answer 14

yw.