Question

17. Divide using synthetic division, and write a summary statement in fraction form.

2x^5 - x^4 +3x^2 - x + 5 over x - 1

2x4 - 3x3 - x + six divided by quantity x minus one
2x4 + x3 - x2 + 2x + 1 + six divided by quantity x minus one
2x4 + x3 + x2 + 4x + 3 + eight divided by quantity x minus one
2x4 + x3 + 4x2 + 3x + eight divided by quantity x minus one

18. State the Vertical Asymptote of the rational function.
f(x) = (x - 6)(x+7) over x^2 - 4

x = 6, x = -7
x = 2, x = -2
None
x = -6, x = 7

19.
State the horizontal asymptote of the rational function.
f (x) = x^2 + 6x - 8 over x - 8.

Answer 1

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

Step-by-step explanation:

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.)