Precalculus : Polynomial Functions

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #10 : Find Intercepts And Asymptotes

Find the slant and vertical asymptotes for the equation

.

Possible Answers:

Vertical asymptote: ; Slant asymptote: 

Vertical asymptote: ; Slant asymptote: 

Vertical asymptote: ; Slant asymptote:

Vertical asymptote: ; Slant asymptote:

Vertical asymptote: ; Slant asymptote: 

Correct answer:

Vertical asymptote: ; Slant asymptote: 

Explanation:

To find the vertical asymptote, just set the denominator equal to 0:

To find the slant asymptote, divide the numerator by the denominator, but ignore any remainder. You can use long division or synthetic division.

               

          

                         

                   

                                       

The slant asymptote is

.

Example Question #31 : Rational Functions

Find the slant asymptote for

.

Possible Answers:

This graph does not have a slant asymptote.

Correct answer:

This graph does not have a slant asymptote.

Explanation:

By factoring the numerator, we see that this equation is equivalent to

.

That means that we can simplify this equation to .

That means that isn't the slant asymptote, but the equation itself. 

is definitely an asymptote, but a vertical asymptote, not a slant asymptote. 

Example Question #32 : Rational Functions

Find the y-intercept of , if any.

Possible Answers:

Correct answer:

Explanation:

Be careful not to confuse this equation with the linear slope-intercept form. The y-intercept of an equation is the y-value when the x-value is zero.

Substitute the value of  into the equation.

Simplify the equation.

The y-intercept is:  

Example Question #33 : Rational Functions

Find the horizontal asymptote of the function:

Possible Answers:

Correct answer:

Explanation:

To find the horizontal asymptote, take the leading term of the numerator and the denominator and divide. In this case:

  

 

Example Question #41 : Rational Functions

Find the vertical and horizontal asymptotes of the function

Possible Answers:

Vertical Asymptotes: 

Horizontal Asymptote: 

Vertical Asymptotes: 

Horizontal Asymptote: 

Vertical Asymptotes: 

Horizontal Asymptote: 

Vertical Asymptotes: 

Horizontal Asymptote: 

Correct answer:

Vertical Asymptotes: 

Horizontal Asymptote: 

Explanation:

The function

is already in simplified form.

To find the vertical asymptotes, we set the denominator equal to  and solve for .

yields the vertical asymptotes

 

To find the horizontal asymptote, we examine the largest degree of  between the numerator and denominator

Note that

Because the largest degree of  in the numerator is less than the largest degree of  in the denominator, or

we find the horizontal symptote to be 

 

Example Question #3 : Asymptotes

Determine the asymptotes, if any:  

Possible Answers:

Correct answer:

Explanation:

Factorize both the numerator and denominator.

Notice that one of the binomials will cancel.

The domain of this equation cannot include .

The simplified equation is:

Since the  term canceled, the  term will have a hole instead of an asymptote.   

Set the denominator equal to zero.

Subtract one from both sides.

There will be an asymptote at only:  

The answer is:  

Example Question #4 : Asymptotes

Where is an asymptote located, if any?   

Possible Answers:

Correct answer:

Explanation:

Factor the numerator and denominator.

Rewrite the equation.

Notice that the  will cancel.  This means that the root of  will be a hole instead of an asymptote.

Set the denominator equal to zero and solve for x.

An asymptote is located at:  

The answer is:  

Example Question #1 : Integral And Rational Zeros Of Polynomial Functions

Use the Rational Zero Theorem to find all potential rational zeros of the polynomial . Which of these is NOT a potential zero?

Possible Answers:

Correct answer:

Explanation:

To find the potential rational zeros by using the Rational Zero Theorem, first list the factors of the leading coefficient and the constant term:

Constant 24: 1, 2, 3, 4, 6, 8, 12, 24

Leading coefficient 2: 1, 2

Now we have to divide every factor from the first list by every factor of the second:

Removing duplicates [for example, and are both equivalent to 1] gives us the following list:

The only choice not on this list is .

Example Question #2 : Integral And Rational Zeros Of Polynomial Functions

Consider the polynomial . Of the potential rational zeros provided by the Rational Zero Theorem, which can we determine to NOT be a solution?

Possible Answers:

Correct answer:

Explanation:

The potential zeros must have a factor of -15 as their numerator and a factor of 6 as their denominator. This eliminates  as a possibility since 6 is not a factor of -15.

Now we need to test which of these values actually give zero when plugged into the polynomial.

First, :

Now :

Finally :

Since this one doesn't give us zero, it is not a solution of the polynomial.

 

 

Example Question #1 : Find The Zeros Of A Function Using The Rational Zeros Theorem

Use Rational Zeros Theorem to find all potential rational zeros of the polynomial . Which of the following sets ONLY includes potential zeros?

Possible Answers:

Correct answer:

Explanation:

To use Rational Zeros Theorem, take all factors of the constant term and all factors of the leading coefficient.  That gives you:

 

Constant Term = 12, so Factors: 1, 2, 3, 4, 6, 12

Leading Coefficient = 3, so Factors: 1, 3

 

Then divide every number from the first list by every number from the second -- keeping in mind that both positive and negative values are possible. Of course, since this is a multiple-choice question you can also use process-of-elimination to get rid of any values that wouldn't be possible from such division. Note that the only prime factors you have are 2 and 3, meaning that there is no way to produce a 5 or 7. That eliminates (or at least makes very suspect) several choices, leaving the choice:

If you perform division from the two lists you should see how you can arrive at each value from this choice: dividing 12 by 3 would give you 4 (or negative 4, again as both + and - values are possible); dividing 2 by 3 would give you , and dividing 2 by 1 or 6 by 3 would give you 2. Therefore this choice only includes possible zeros.

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