SOLUTION: Given p(x) = x^3+5x^2+4x/3(x+1)(x^2−7x+12), find: (a) X-intercept(s) (b) Y-intercept (c) Vertical asymptote(s) (d) Hole(s) (e) End behavior / horizontal asymptote

Question 1167485: Given p(x) = x^3+5x^2+4x/3(x+1)(x^2−7x+12),
find:
(a) X-intercept(s)
(b) Y-intercept
(c) Vertical asymptote(s)
(d) Hole(s)
(e) End behavior / horizontal asymptote
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

-intercepts are the points

where each

is a real zero of the numerator function. The

-intercept is the point

where

and

is the numerator function. There is a vertical asymptote at the line

for each factor

of the denominator function that is not also a factor of the numerator function. There is a hole where ever a linear factor of the denominator has a common linear factor in the numerator.

Horizontal asymptotes:

Case 1: The degree of the numerator polynomial is smaller than the degree of the denominator polynomial -- the line

(i.e. the

-axis) is the horizontal asymptote.

Case 2: The degree of the numerator polynomial is equal to the degree of the denominator polynomial -- the line

where

is the lead coefficient of the numerator polynomial and

is the lead coefficient of the denominator polynomial.

Case 3: The degree of numerator polynomial is greater than the degree of the denominator polynomial -- there is no horizontal asymptote, rather there is a slant asymptote. The RHS of the equation of the slant asymptote is the quotient obtained from performing polynomial long division excluding any remainder.

John

My calculator said it, I believe it, that settles it

I > Ø

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