17. 2x^4 + x^3 + x^2 + 4x + 3 + 8/(x -1)
18. x = 2, x = -2
19. The slant asymptote is y = x + 14
<h3>Explanation:</h3>17. See the first attachment for the synthetic division. The summary statement is the expression represented by the result.
18. The denominator is the difference of two squares, so is readily factored to ...
... (x -2)(x +2)
The zeros of this product are the locations of the vertical asymptotes of f(x). They are ...
... x = 2, x = -2.
19. Dividing the numerator by the denominator (using synthetic division, if you like) gives the result ...
... f(x) = x + 14 + 104/(x -8)
The linear expression y=x+14 defines the end behavior when x gets large. That is, it is the slant asymptote of the function. See the second attachment for a graph.
(There will be a horizontal asymptote only when the degrees of numerator and denominator are the same. In this case, they are not.)
