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 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

The polynomial y = x^3 - x ^ 2 - x - 15 intersects the x-axis at point (3,0) . Find the other two solutions.

Possible Answers:

$1 \pm 4i$

$-1 \pm 2i$

$- 1 \pm 4i$

$-1 \pm 5 i$

$-2 \pm 4i$

Correct answer:

$-1 \pm 2i$

Explanation:

Since we know that one of the zeros of this polynomial is 3, we know that one of the factors is x - 3 . To find the other two zeros, we can divide the original polynomial by x - 3 , either with long division or with synthetic division:

$\begin{matrix} \underline{3}| \enspace 1 \enspace -1 \enspace -1 \enspace -15 \\\underline{ + \enspace \enspace \enspace \enspace 3 \enspace \enspace \enspace \enspace 6 \enspace \enspace \enspace 15 }\\ 1 \enspace \enspace \enspace 2 \enspace \enspace \enspace 5 \enspace \enspace \enspace 0 \end{matrix}$

This gives us the second factor of 1x^2 + 2 x + 5 . We can get our solutions by using the quadratic formula:

$\\x = \frac{-2 \pm \sqrt{4 - 4(1)(5)}}{2} \\ \\= \frac{-2 \pm \sqrt{ 4 - 20 }}{2 } \\ \\= \frac{-2 \pm \sqrt{-16}}{2} \\ \\= \frac{-2 \pm 4i }{2}\\ \\ = -1 \pm 2i$

Report an Error

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

Find all the real and complex zeroes of the following equation: 2x^5+x^4-2x-1

Possible Answers:

$x=1, \frac{-1}{2}$

$x=1, -1, \frac{-1}{2}, \pm i$

$x=1, -1, \frac{-1}{2}, i$

$x=1,\frac{-1}{2}, i$

$x=1, \frac{-1}{2}, \pm i$

Correct answer:

$x=1, -1, \frac{-1}{2}, \pm i$

Explanation:

First, factorize the equation using grouping of common terms:

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

Next, setting each expression in parenthesis equal to zero yields the answers.

$(x+1)=0\rightarrow x=-1$ $(x-1)=0\rightarrow x=1$ $(x^2+1)=0\rightarrow x=\pm i$

$(2x+1)=0\rightarrow x=\frac{-1}{2}$

Report an Error

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

Find all the zeroes of the following equation and their multiplicity: g(t)=t^5-6t^3+9t

Possible Answers:

$t=0, \pm \sqrt{3i}$ (multiplicity of 2 on 0, multiplicity of 1 on $\pm \sqrt{3i}$

$t=0, \pm \sqrt{3i}$ (multiplicity of 1 on 0, multiplicity of 2 on $\pm \sqrt{3i}$

$t=0, \pm \sqrt{3}$ (multiplicity of 1 on 0, multiplicity of 2 on $\pm \sqrt{3}$

$t=0, \pm \sqrt{3}$ (multiplicity of 2 on 0, multiplicity of 1 on $\pm \sqrt{3}$

Correct answer:

$t=0, \pm \sqrt{3}$ (multiplicity of 1 on 0, multiplicity of 2 on $\pm \sqrt{3}$

Explanation:

First, pull out the common t and then factorize using quadratic factoring rules:

This equation has solutions at two values: when t=0 and when

$(t^2-3)^2=0\rightarrow t=\pm \sqrt{3}$ Therefore, But since the degree on the former equation is one and the degree on the latter equation is two, the multiplicities are 1 and 2 respectively.

Report an Error

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

Find a fourth degree polynomial whose zeroes are -2, 5, and $-2\pm \sqrt{2i}$

Possible Answers:

x^4-16x^2-58x-60

x^3-16x^2-58x-60

x^4-x^3-16x^2-58x-60

x^4+x^3+16x^2+58x-60

x^4+x^3-16x^2-58x-60

Correct answer:

x^4+x^3-16x^2-58x-60

Explanation:

This one is a bit of a journey. The expressions for the first two zeroes are easily calculated, (x+2) and (x-5) respectively. The last expression must be broken up into two equations:

which are then set equal to zero to yield the expressions $(x+2-i\sqrt{2})$ and $(x+2+i\sqrt{2})$

Finally, we multiply together all of the parenthesized expressions, which multiplies out to x^4+x^3-16x^2-58x-60=0

Report an Error

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

The third degree polynomial expression f(x)=x^3-x^2+4x-4 has a real zero at x=1 . Find all of the complex zeroes.

Possible Answers:

$x=\pm \sqrt{2}$

$x=\pm 2i$

$x=2\pm \sqrt{2i}$

$x=\pm 4i$

$x=2\pm 2i$

Correct answer:

$x=\pm 2i$

Explanation:

First, factor the expression by grouping:

$x^3-x^2+4x-4\rightarrow x^2(x-1)+4(x-1)\rightarrow (x^2+4)\cdot(x-1)$

To find the complex zeroes, set the term (x^2+4) equal to zero:

$x^2=-4\rightarrow x=\pm \sqrt{-2}=\pm 2i$

Report an Error

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

Find all the real and complex zeros of the following equation: $2x^{5}+x^{4}-2x-1$

Possible Answers:

$x=1,-1,\frac{-1}{2},i$

$x=1,\frac{-1}{2},i$

$x=1, \frac{-1}{2},\pm i$

$x=1,\frac{-1}{2}$

$x=1, -1, \frac{-1}{2},\pm i$

Correct answer:

$x=1, -1, \frac{-1}{2},\pm i$

Explanation:

First, factorize the equation using grouping of common terms:

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

Next, setting each expression in parentheses equal to zero yields the answers.

$(x+1)=0\rightarrow x=-1(x-1)=0\rightarrow x=1(x^{2}+1)=0\rightarrow x=\pm i{(2x+1)=0\rightarrow x=\frac{-1}{2}}$

Report an Error

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

Find all the zeroes of the following equation and their multiplicity: $g(t)=t^{5}-6t^{3}+9t$

Possible Answers:

$t=0,\pm \sqrt{3}$ (multiplicity of 2 on 0, multiplicity of 1 on $\pm \sqrt{3}$ )

$t=0,\pm \sqrt{3}$ (multiplicity of 1 on 0, multiplicity of 2 on $\pm \sqrt{3}$ )

$t=0,\pm \sqrt{3i}$ (multiplicity of 2 on 0, multiplicity of 1 on $\pm \sqrt{3i}$ )

$t=0,\pm \sqrt{3i}$ (multiplicity of 1 on 0, multiplicity of 2 on $\pm \sqrt{3i}$ )

Correct answer:

$t=0,\pm \sqrt{3}$ (multiplicity of 1 on 0, multiplicity of 2 on $\pm \sqrt{3}$ )

Explanation:

First, pull out the common t and then factorize using quadratic factoring rules: $t^{5}-6t^{3}+9t\rightarrow t(t^{4}-6t^{2}+9)\rightarrow t(t^{2}-3)^{2}$

This equation has a solution as two values: when t=0 , and when $\left ( t^{2}-3 \right )^{2}=0\rightarrow t=\pm \sqrt{3}$ . Therefore, But since the degree on the former equation is one and the degree on the latter equation is two, the multiplicities are 1 and 2 respectively.

Report an Error

Example Question #431 : Pre Calculus

Find a fourth-degree polynomial whose zeroes are -2,5 , and $-2\pm \sqrt{2i}$

Possible Answers:

$x^{4}+x^{3}+16x^{2}+58x-60$

$x^{4}-x^{3}-16x^{2}-58x-60$

$x^{3}-16x^{2}-58x-60$

$x^{4}-16x^{2}-58x-60$

$x^{4}+x^{3}-16x^{2}-58x-60$

Correct answer:

$x^{4}+x^{3}-16x^{2}-58x-60$

Explanation:

This one is a bit of a journey. The expressions for the first two zeroes are easily calculated, $\left ( x+2 \right )$ and $\left ( x-5 \right )$ respectively. The last expression must be broken up into two equations: $x=-2-i\sqrt{2},x=-2+i\sqrt{2}$ which are then set equal to zero to yield the expressions $\left ( x+2-i\sqrt{2} \right )$ and $\left ( x+2+i\sqrt{2} \right )$

Finally, we multiply together all of the parenthesized expressions, which multiplies out to $x^{4}+x^{3}-16x^{2}-58x-60=0$

Report an Error

Example Question #432 : Pre Calculus

The third-degree polynomial expression $f(x)=x^{3}-x^{2}+4x-4$ has a real zero at x=1 . Find all of the complex zeroes.

Possible Answers:

Correct answer:

Explanation:

First, factor the expression by grouping: $x^{3}-x^{2}+4x-4\rightarrow x^{2}(x-1)+4(x-1)\rightarrow (x^{2}+4)\cdot (x-1)$

To find the complex zeroes, set the term $(x^{2}+4)$ equal to zero: $x^{2}=-4\rightarrow x=\pm \sqrt{-2}=\pm 2i$

Report an Error

Example Question #1 : Find The Sum And Product Of The Zeros Of A Polynomial

Given y=5x^2+14x-3 , determine the sum and product of the zeros respectively.

Possible Answers:

$\frac{14}{5},-\frac{3}{5}$

$\frac{16}{5}, -\frac{3}{5}$

$-\frac{14}{5},-\frac{3}{5}$

$\frac{1}{5},-3$

$\frac{7}{5},-\frac{21}{5}$

Correct answer:

$-\frac{14}{5},-\frac{3}{5}$

Explanation:

To determine zeros of y=5x^2+14x-3 , factorize the polynomial.

5x^2+14x-3=(5x-1)(x+3)

Set each of the factorized components equal to zero and solve for .

5x-1=0

$x=\frac{1}{5}$

x+3=0

x=-3

The sum of the roots:

$\frac{1}{5}+(-3)= \frac{1}{5}-\frac{15}{5}= -\frac{14}{5}$

The product of the roots:

$(\frac{1}{5})(-3)= -\frac{3}{5}$

Report an Error

View Pre-Calculus Tutors

Alejandro
Certified Tutor

Kennesaw State University, Bachelor of Science, Computer Software Engineering.

View Pre-Calculus Tutors

Charles
Certified Tutor

Pennsylvania State University-Main Campus, Bachelor of Science, Mathematics.

View Pre-Calculus Tutors

Kenneth
Certified Tutor

North Georgia University, Bachelors, Accounting. Brenau University, Masters, Business.

All Precalculus Resources

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

Popular Subjects

SSAT Tutors in Miami, Spanish Tutors in Houston, French Tutors in Los Angeles, Reading Tutors in Boston, ACT Tutors in Washington DC, SAT Tutors in Seattle, Statistics Tutors in Philadelphia, Math Tutors in Denver, Statistics Tutors in Washington DC, MCAT Tutors in Atlanta

Popular Courses & Classes

SAT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in New York City, ACT Courses & Classes in Denver, SSAT Courses & Classes in Seattle, ISEE Courses & Classes in San Diego, LSAT Courses & Classes in Seattle, ACT Courses & Classes in Seattle, ISEE Courses & Classes in Denver, Spanish Courses & Classes in Phoenix, MCAT Courses & Classes in Philadelphia

Popular Test Prep

GMAT Test Prep in San Francisco-Bay Area, GMAT Test Prep in Washington DC, ACT Test Prep in Seattle, LSAT Test Prep in Miami, ACT Test Prep in Denver, ISEE Test Prep in San Francisco-Bay Area, ISEE Test Prep in Dallas Fort Worth, LSAT Test Prep in San Diego, MCAT Test Prep in Seattle, GRE Test Prep in Atlanta

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 : Find Complex Zeros Of A Polynomial Using The Fundamental Theorem Of Algebra

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

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

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

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

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

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

Example Question #431 : Pre Calculus

Example Question #432 : Pre Calculus

Example Question #1 : Find The Sum And Product Of The Zeros Of A Polynomial

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: