Michael J. answered • 04/04/17

Tutor

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Mastery of Limits, Derivatives, and Integration Techniques

First, we need to factor this function. We can try factoring by grouping.

x

^{4}- x - 2x^{3}+ 2______________ =

x

^{3}- 4xx(x

^{3}- 1) - 2(x^{3}- 1)_________________ =

x(x

^{2}- 4)(x - 2)(x

^{3}- 1)______________ =

x(x - 2)(x + 2)

(x - 2)(x - 1)(x

^{2}+ x + 1)_____________________ =

x(x - 2)(x + 2)

We can cancel out (x - 2). This means that x=2 is a hole in the graph.

(x - 1)(x

^{2}+ x + 1)________________ =

x(x + 2)

Now that we factored the function, we can find all the intercepts and asymptotes.

To find the x-intercepts, set the numerator equal to zero and solve for x. Since the quadratic factor gives you complex roots, the only x-intercept is x=1.

To find y-intercept, set x=0 and evaluate the function.

To find the vertical asymptote, set the denominator equal to zero and solve for x. Your vertical asymptotes will be x=0 and x=0 and x=-2

If we expand the simplified expression, we can see that the degree of the numerator is greater than the degree of the denonomintor. Therefore, you will have a slant asymptote.

To find the slant asymptote, expand the simplified expression. Divide the expanded numerator by the expanded denominator using long division process. You will have a quotient, and remainder.

The equation of the slant asymptote is the quotient part in the division.

So lets review.

x-intercept = (1, 0)

Hole at x=2

Y-intercept: none

vertical asymptote: x = 0 and x = -2

Slant asymptote: Quotient part using long division

You now have all the information. Use them to sketch an accurate curve.

John C.

04/05/17