Your function "x^3+5x^2+4x/3(x+1)(x^2−7x+12)" means this:
If you are working on a problem like this, you should know that parentheses in the right places are important....
Undoubtedly the function is supposed to be "(x^3+5x^2+4x)/(3(x+1)(x^2-7x+12))"
Now we can answer all the questions quickly by factoring numerator and denominator:
(a) x-intercept(s): wherever there is a linear factor in the numerator without a like factor in the denominator. (x=0, x=-4)
(b) y-intercept: (0,b) where b = p(0). Since x is a factor of the numerator, the y-intercept is (0,0).
(c) Vertical asymptote(s): wherever there is a linear factor in the denominator without a like factor in the numerator. (x=3, x=4)
(d) Hole(s): wherever there are like linear factors in both numerator and denominator. (x=-1)
(e) End behavior/horizontal asymptote: The degrees of numerator and denominator are the same, so the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator. (y = 1/3)
A graph showing the x- and y-intercepts....
Another showing the horizontal asymptote y=1/3....
And one showing the effect of the vertical asymptotes at x=3 and x=4....
And finally one showing the hole at (-1,-0.5)....
The hole will show up better on a graphing utility like a TI83 calculator...
