We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Polynomial Functions

Study concepts, example questions & explanations for Precalculus

Create An Account

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Express A Polynomial As A Product Of Linear Factors

Factorize the following expression completely to its linear factors: f(x)=10x^2-4x+15xy-6y

Possible Answers:

$(2x+3)\cdot(5x-2)$

$(2x+3y)\cdot(5x-2)$

$(2x-3y\cdot(5x+2)$

$(2x+3y\cdot(5x+2)$

Correct answer:

$(2x+3y)\cdot(5x-2)$

Explanation:

Use the grouping method to factorize common terms:

$10x^2-4x+15xy-6y\rightarrow 2x(5x-2)+3y(5x-2)\rightarrow(2x+3y)\cdot(5x-2)$

Report an Error

Example Question #1 : Express A Polynomial As A Product Of Linear Factors

Factorize the following expression completely to its linear factors: f(x)=

2x^2+3xy-10x-15y

Possible Answers:

$(x-5)\cdot(x+y)$

$(x+5)\cdot(2x+y)$

$(x-5)\cdot(2x-y)$

$(x-5)\cdot(2x+3y)$

$(x+5)\cdot(2x+y)$

Correct answer:

$(x-5)\cdot(2x+3y)$

Explanation:

Use grouping method to factorize common terms:

$2x^2+3xy-10x-15y\rightarrow x(2x+3y)-5(2x+3y)\rightarrow (x-5)\cdot(2x+3y)$

Report an Error

Example Question #1 : Express A Polynomial As A Product Of Linear Factors

Express the polynomial as a product of linear factors: 2x^3 + 4x^2 - 2x - 4

Possible Answers:

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

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

2(x-1)^2(x+2)

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

2(x^2+1)(x+2)

Correct answer:

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

Explanation:

First pull out the common factor of 2, and then factorize:

Report an Error

Example Question #1 : Express A Polynomial As A Product Of Linear Factors

Find the real zeros of the equation using factorization: f(x)= 5x^6-5x^4+x^2-1

Possible Answers:

x=-1

x=1, -1

x=1

There are no real zeros.

x=0

Correct answer:

x=1, -1

Explanation:

Use grouping to factorize the common terms:

$5x^6-5x^4+x^2-1\rightarrow 5x^4(x^2-1)+1(x^2-1)\rightarrow (5x^4+1)(x^2-1)\rightarrow (5x^4+1)(x+1)(x-1)$

Report an Error

Example Question #10 : Express A Polynomial As A Product Of Linear Factors

Express the following polynomial as a product of linear factors: $2x^{3}+4x^{2}-2x-4$

Possible Answers:

$2(x^{2}+1)(x+2)$

$2(x-1)^{2}(x+2)$

Correct answer:

Explanation:

First, pull out the common factor of , and then factorize:

$2(x^{3}+2x^{2}-x-2)\rightarrow 2[x^{2}(x+2)-1(x+2)]\rightarrow 2(x^{2}-1)(x+2)\rightarrow 2(x-1)(x+1)(x+2)$

Report an Error

Example Question #11 : Express A Polynomial As A Product Of Linear Factors

Factorize the following expression completely to its linear factors: $f(x)=2x^{2}+3xy-10x-15y$

Possible Answers:

$\left ( x-5 \right )\cdot \left ( x+y \right )$

$(x-5)\cdot (2x-y)$

$(x+5)\cdot (2x+y)$

$\left ( x+5 \right )\cdot \left ( 2x+y \right )$

$(x-5)\cdot (2x+3y)$

Correct answer:

$(x-5)\cdot (2x+3y)$

Explanation:

Use the grouping method to factorize common terms: $2x^{2}+3xy-10x-15y\rightarrow x(2x+3y)-5(2x+3y)\rightarrow (x-5\)\cdot \left ( 2x+3y \right )$

Report an Error

Example Question #11 : Express A Polynomial As A Product Of Linear Factors

Factorize the following polynomial expression completely to its linear factors: $f(x)=10x^{2}-4x+15xy-6y$

Possible Answers:

$(2x+3)\cdot (5x-2)$

$(2x+3y\cdot (5x+2)$

$(2x+3y)\cdot (5x-2)$

$(2x-3y\cdot (5x+2)$

Correct answer:

$(2x+3y)\cdot (5x-2)$

Explanation:

Use the grouping method to factorize common terms: $10x^{2}-4x+15xy-6y\rightarrow 2x(5x-2)\rightarrow (2x+3y)\cdot (5x-2)$

Report an Error

Example Question #1 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

What are the roots of

$\small f(x)=x^3-x^2+2x-2$

including complex roots, if they exist?

Possible Answers:

$\small x=1, \sqrt{2}i, -\sqrt{2}i$

$\small x=1,-1,\sqrt{2}$

$\small x=1$

$\small x=1, \sqrt{2}, -\sqrt{2}$

Correct answer:

$\small x=1, \sqrt{2}i, -\sqrt{2}i$

Explanation:

One of the roots is $\small x=1$ because if we plug in 1, we get 0. We can factor the polynomial as

$\small \small f(x)=x^3-x^2+2x-2=(x-1)(x^2+2)$

So now we solve the roots of $\small x^2+2$ .

$x^2=-2 \rightarrow x= \pm\sqrt{-2}= \pm i\sqrt2$

The root will not be real.

The roots of this polynomial are $\small x=\sqrt{2}i, -\sqrt{2}i$ .

So, the roots are $\small x=1, \sqrt{2}i, -\sqrt{2}i$

Report an Error

Example Question #2 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

The polynomial y = 2x^3 - 7 x ^ 2 + 58 x - 78 has a real zero at 1.5. Find the other two zeros.

Possible Answers:

$x = 2 \pm 5 i$

x = 6, -4

$x = 1 \pm \frac{\sqrt{108}}{2}$

$x = 1 \pm 5i$

$x = 1 \pm 10 i$

Correct answer:

$x = 1 \pm 5i$

Explanation:

If this polynomial has a real zero at 1.5, that means that the polynomial has a factor that when set equal to zero has a solution of x = 1.5 . We can figure out what this is this way:

x - 1.5 = 0 multiply both sides by 2

2x - 3 = 0 is the factor

Now that we have one factor, we can divide to find the other two solutions:

$\begin{matrix} x^2 - 2x + 26\\ 2x - 3 \overline{| 2x^3 - 7 x^2 + 58 x - 78} \\ \underline{2x^3 - 3x^2} \enspace \enspace \enspace \enspace \\ \enspace \enspace \enspace \enspace -4x^2 + 58 x \\ \enspace \enspace \enspace \enspace \underline{-4x^2 + 6x} \\ \enspace \enspace \enspace \enspace \enspace \enspace \enspace52x - 78 \\ \enspace \enspace \enspace \enspace \enspace \enspace \enspace \underline{52x - 78} \\ \enspace \enspace \enspace \enspace \enspace \enspace \enspace 0 \end{matrix}$

To finish solving, we can use the quadratic formula with the resulting quadratic, x^2 - 2x + 26 :

$x = \frac{2 \pm \sqrt {4 - 4(1)(26)}}{2(1)} = \frac{2 \pm \sqrt{-100}}{2}= \frac{2 \pm 10i }{2 } = 1 \pm 5i$

Report an Error

Example Question #3 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

If the real zero of the polynomial y = x^3 + 5 x ^2 + x - 75 is 3, what are the complex zeros?

Possible Answers:

$8 \pm 5i$

$-8 \pm 6i$

$-4 \pm 4i$

$4 \pm 4i$

$-4 \pm 3i$

Correct answer:

$-4 \pm 3i$

Explanation:

We know that the real zero of this polynomial is 3, so one of the factors must be x - 3 . To find the other factors, we can divide the original polynomial by x - 3 , either by long division or synthetic division:

$\begin{matrix} \underline{3}| \enspace 1 \enspace 5 \enspace 1 \enspace -75 \\\underline{ + \enspace \enspace \enspace 3 \enspace 24 \enspace \enspace 75 }\\ \enspace \enspace 1 \enspace 8 \enspace 25 \enspace \enspace \enspace 0 \end{matrix}$

This gives us a second factor of x^2 + 8 x + 25 which we can solve using the quadratic formula:

$x = \frac{-8 \pm \sqrt{64 - 4(1)(25)}}{2} = \frac{-8 \pm \sqrt{-36}}{ 2 } = \frac{-8 \pm 6i }{2} = -4 \pm 3i$

Report an Error

View Pre-Calculus Tutors

Steven
Certified Tutor

University of Louisville, Bachelor of Science, Mechanical Engineering. University of Phoenix-Louisville Campus, Master of Eng...

View Pre-Calculus Tutors

Leonard
Certified Tutor

Florida State University, Bachelor of Science, Environmental Science.

View Pre-Calculus Tutors

Valsalakumari
Certified Tutor

Texas A & M University-Kingsville, Bachelors, Mathematics. Texas A & M University-Kingsville, Masters, Mathematics.

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Calculus Tutors in Seattle, French Tutors in Miami, Algebra Tutors in Los Angeles, GRE Tutors in Phoenix, Biology Tutors in Washington DC, ISEE Tutors in Phoenix, Math Tutors in Miami, Calculus Tutors in Philadelphia, French Tutors in Houston, GRE Tutors in Boston

Popular Courses & Classes

ACT Courses & Classes in Washington DC, Spanish Courses & Classes in Chicago, GMAT Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in San Diego, GRE Courses & Classes in Philadelphia, GMAT Courses & Classes in Denver, ISEE Courses & Classes in Denver, Spanish Courses & Classes in Los Angeles, SSAT Courses & Classes in Los Angeles, SSAT Courses & Classes in New York City

Popular Test Prep

SAT Test Prep in Phoenix, ISEE Test Prep in Seattle, MCAT Test Prep in New York City, GRE Test Prep in Seattle, ACT Test Prep in Seattle, SSAT Test Prep in Denver, ACT Test Prep in Atlanta, SSAT Test Prep in San Francisco-Bay Area, SSAT Test Prep in Los Angeles, LSAT Test Prep in Dallas Fort Worth

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Precalculus : Polynomial Functions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Express A Polynomial As A Product Of Linear Factors

Example Question #1 : Express A Polynomial As A Product Of Linear Factors

Example Question #1 : Express A Polynomial As A Product Of Linear Factors

Example Question #1 : Express A Polynomial As A Product Of Linear Factors

Example Question #10 : Express A Polynomial As A Product Of Linear Factors

Example Question #11 : Express A Polynomial As A Product Of Linear Factors

Example Question #11 : Express A Polynomial As A Product Of Linear Factors

Example Question #1 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

Example Question #2 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

Example Question #3 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: