For the following rational function answer the following que

For the following rational function, answer the following questions: r(x) = x^3 + 3x^2 + 2x/x^2 - x - 2 Find the domain of r(x) (write in interval notation):________ Reduce r(x) to lowest terms, if possible:________ Find the coordinates of the x-intercept(s):________ Find the coordinates of the y-intercept:________

Solution

r(x) = (x^3 +3x^2 +2x)/(x^2 - x-2)

a) Domain: Den, should not be zero:

x^2 -x -2 =0 ---> x^2 -2x +x -2 =0

x(x-2) +1(x-2) =0

(x+1)(x-2) =0

x= -1 ; x= 2

Domain: ( -inf, -1) U ( -1 , 2) U ( 2, inf)

b) r(x) = (x^3 +3x^2 +2x)/(x^2 - x-2)

= x(x^2 + 3x +2)/(x+1)(x-2)

=x(x^2+ 2x +x +2)/(x+1)(x-2)

= x(x+1)(x+2)/(x+1)(x-2)

= x(x+2)/(x-2)

c) X intercepts: x=0 , x=-2

d) y intercept : plug x=0 in r(x)

r(0) = 0

So, r(x) passses through origin