I need help with this Polynomial and Rational Functions Question.

Question

1. Identify any asymptote(s) and find the location of any hole(s) of the function. If they do not exist, write NONE and write any asymptotes as equations of lines. 

R(x)=(2x^3+8x^2+6x)/(x^2+5x+6)

Answer 1

Hello Jessica,

For the numerator, I am going to factor as

2x³ + 8x² + 6x

= 2x(x² + 4x + 3)

= 2x(x + 1)(x + 3)

For the denominator, we get

x²+5x+6

= (x + 2)(x + 3),

Upon cancelling the common factors, (x+3), we now have

[2x(x + 1)] / [(x + 2)]

The hole occurs at x = -3, since the (x + 3) factors cancel.

To find the corresponding 'y' value, plug in - 3 into the simplified form, i.e.

2(-3)(-3 + 1) / (-3 + 2)

= (-6)(-2) / (-1)

= -12

→ hole at (-3 , -12)

This leaves the quantity (x + 2) in the denominator

→ Vertical Asymptote at x = -2

There is NO Horizontal Asymptote,

since the degree of the numerator is larger than that of the denominator.

Since the degree of the numerator is ONE larger that that of the denominator,

there is a Slant Asymptote.

You can find this by long dividing R(x) to get

y = 2x - 2 (you only care about the quotient, not the remainder).

Thank you for the question, and have a great day.

Michael E.