Precalculus : Polynomial Functions

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #4 : Integral And Rational Zeros Of Polynomial Functions

Which of the following CANNOT be a zero of the polynomial  according to Rational Zeros Theorem?

Possible Answers:

Correct answer:

Explanation:

To use Rational Zeros Theorem, express a polynomial in descending order of its exponents (starting with the biggest exponent and working to the smallest), and then take the constant term (here that's 6) and the coefficient of the leading exponent (here that's 4) and express their factors:

Constant: 6 has as factors 1, 2, 3, and 6

Coefficient: 4 has as factors 1, 2, and 4

 

Then all possible rational zeros must be formed by dividing a factor from the Constant list by a factor from the Coefficient list (and note that the results should be considered as both positive and negative values).

 

Here if you then look at the answer choices, note that since the divisor - which must be a factor from the Coefficient list of 1, 2, and 4 - cannot then be a multiple of 3, you know that  is not a possible zero of this polynomial

Example Question #5 : Integral And Rational Zeros Of Polynomial Functions

Using Rational Zeros Theorem, determine which of the following is NOT a zero of the polynomial .

Possible Answers:

Correct answer:

Explanation:

To apply Rational Zero Theorem, first organize a polynomial in descending order of its exponents.  Then take the constant term and the coefficient of the highest-valued exponent and list their factors:

Constant: 2 has factors of 1 and 2

Coefficient: 2 has factors of 1 and 2

Then the potential rational zeros need to be formed by dividing a factor from the Constant list by a factor from the Coefficient list. Here you only have 1s and 2s, so your options are 1, 2, and 1/2 (and note that both positive and negative values will work). There is no way to get to 1/4, meaning that that is the correct answer.

Example Question #3 : Graphs Of Polynomial Functions

 

Which of the following is an accurate graph of ?

Possible Answers:

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Varsity1

Correct answer:

Varsity1

Explanation:

is a parabola, because of the general  structure.  The parabola opens downward because .  

Solving tells the x-value of the x-axis intercept;

The resulting x-axis intercept is: .

Setting  tells the y-value of the y-axis intercept;

    

    

    

The resulting y-axis intercept is:

Example Question #1 : How To Graph An Exponential Function

Give the -intercept of the graph of the function

Round to the nearest tenth, if applicable.

Possible Answers:

The graph has no -interceptx

Correct answer:

Explanation:

The -intercept is , where :

The -intercept is .

Example Question #4 : Graphs Of Polynomial Functions

Graph the following function and identify the zeros.

Possible Answers:

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Correct answer:

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Explanation:

This question tests one's ability to graph a polynomial function.

For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7). 

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use algebraic technique to factor the function.

Separating the function into two parts...

Factoring a negative one from the second set results in...

Factoring out  from the first set results in...

The new factored form of the function is,

.

Now, recognize that the first binomial is a perfect square for which the following formula can be used

since 

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

Step 3: Create a table of  pairs.

The values in the table are found by substituting in the x values into the function as follows.

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

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Example Question #5 : Graphs Of Polynomial Functions

Graph the function and identify the roots.

Possible Answers:

Question2

Question3

Question6

Question5

Question12

Correct answer:

Question12

Explanation:

This question tests one's ability to graph a polynomial function.

For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7). 

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use algebraic technique to factor the function.

Recognize that the binomial is a perfect square for which the following formula can be used

since 

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

        

Step 3: Create a table of  pairs.

The values in the table are found by substituting in the x values into the function as follows.

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

Question12

Example Question #6 : Graphs Of Polynomial Functions

Graph the function and identify its roots.

Possible Answers:

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Question5

Question4

Question3

Question6

Correct answer:

Question6

Explanation:

This question tests one's ability to graph a polynomial function.

For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7). 

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use algebraic technique to factor the function.

Recognize that the binomial is a perfect square for which the following formula can be used

since 

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

        

Step 3: Create a table of  pairs.

The values in the table are found by substituting in the x values into the function as follows.

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

Question6

Example Question #7 : Graphs Of Polynomial Functions

Graph the function and identify its roots.

Possible Answers:

Question4

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Question2

Question3

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Correct answer:

Question4

Explanation:

This question tests one's ability to graph a polynomial function.

For the purpose of Common Core Standards, "graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior" falls within the Cluster C of "Analyze Functions Using Different Representations" concept (CCSS.MATH.CONTENT.HSF-IF.C.7). 

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Use algebraic technique to factor the function.

Recognize that the binomial is a perfect square for which the following formula can be used

since 

thus the simplified, factored form is,

.

Step 2: Identify the roots of the function.

To find the roots of a function set its factored form equal to zero and solve for the possible x values.

        

Step 3: Create a table of  pairs.

The values in the table are found by substituting in the x values into the function as follows.

Step 4: Plot the points on a coordinate grid and connect them with a smooth curve.

Question4

Example Question #1 : Write The Equation Of A Polynomial Function Based On Its Graph

Which could be the equation for this graph?

Polynomial

Possible Answers:

Correct answer:

Explanation:

This graph has zeros at 3, -2, and -4.5. This means that , , and . That last root is easier to work with if we consider it as and simplify it to . Also, this is a negative polynomial, because it is decreasing, increasing, decreasing and not the other way around.

Our equation results from multiplying , which results in .

Example Question #2 : Write The Equation Of A Polynomial Function Based On Its Graph

Write the quadratic function for the graph:

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Possible Answers:

Correct answer:

Explanation:

Because there are no x-intercepts, use the form , where vertex  is , so , , which gives

          

          

          

          

      

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