\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Precalculus - Rational Roots Test

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

Rational Roots Test

The Rational Roots Test is quick way to get a list of all possible rational roots to a polynomial. It is a very good place to start when trying to find roots of higher order polynomials (3 or higher). Once you have a rational root or two, you can reduce the order of the polynomial using synthetic division to find irrational and complex roots.

However, that said, we never used this technique in calculus, so as far as we are concerned, you do not need to know this for calculus. We include this page mostly for those of you college algebra that may be part of your curriculum. However, as usual, check with your instructor to see what they require.
If you need to know this, we recommend this video explaining this technique and showing an example and we include some practice problems.

PatrickJMT - Rational Roots Test [6min-50secs]

video by PatrickJMT

Pre-calculus Demystified, Second Edition

Practice

Unless otherwise instructed, find all possible rational roots of these polynomial functions using the rational roots theorem. Determine which are actual roots and factor the polynomial as much as possible, using the roots.

\( f(x) = 3x^3 - 4x^2 - 17x +6 \)

Problem Statement

Find all possible rational roots of \( f(x) = 3x^3 - 4x^2 - 17x +6 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

Brian McLogan - 4185 video solution

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

\( f(x) = x^3 + 2x^2 - 5x - 6 \)

Problem Statement

Find all possible rational roots of \( f(x) = x^3 + 2x^2 - 5x - 6 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

The Organic Chemistry Tutor - 4186 video solution

Log in to rate this practice problem and to see it's current rating.

\( x^3 + 8x^2 + 11x - 20 \)

Problem Statement

Find all possible rational roots of \( x^3 + 8x^2 + 11x - 20 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

The Organic Chemistry Tutor - 4187 video solution

Log in to rate this practice problem and to see it's current rating.

\( x^3 - 11x + 6 \)

Problem Statement

Find all possible rational roots of \( x^3 - 11x + 6 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

The Organic Chemistry Tutor - 4188 video solution

Log in to rate this practice problem and to see it's current rating.

\( f(x) = x^3 - 6x^2 + 11x - 6 \)

Problem Statement

Find all possible rational roots of \( f(x) = x^3 - 6x^2 + 11x - 6 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

Brian McLogan - 4198 video solution

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

\( f(x) = 2x^3 - 12x^2 + 22x - 12 \)

Problem Statement

Find all possible rational roots of \( f(x) = 2x^3 - 12x^2 + 22x - 12 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

Mario's Math Tutoring - 4189 video solution

Log in to rate this practice problem and to see it's current rating.

\( f(x) = 2x^3 - 3x^2 - 11x + 6 \)

Problem Statement

Find all possible rational roots of \( f(x) = 2x^3 - 3x^2 - 11x + 6 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

PatrickJMT - 4184 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

\( f(x) = 6x^3 - 11x^2 - 26x + 15 \)

Problem Statement

Find all possible rational roots of \( f(x) = 6x^3 - 11x^2 - 26x + 15 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

MIP4U - 4191 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

\( p(x) = 3x^3 - 5x^2 -14x - 12 \)

Problem Statement

Find all possible rational roots of \( p(x) = 3x^3 - 5x^2 -14x - 12 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

Brian McLogan - 4192 video solution

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

\( f(x) = 2x^3 + 5x^2 - x - 6 \)

Problem Statement

Find all possible rational roots of \( f(x) = 2x^3 + 5x^2 - x - 6 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

Brian McLogan - 4193 video solution

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

\( f(x) = x^4 - x^3 - 7x^2 + 5x + 10 \)

Problem Statement

Find all possible rational roots of \( f(x) = x^4 - x^3 - 7x^2 + 5x + 10 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

Brian McLogan - 4195 video solution

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

\( -4x^3 + x^2 -3x + 10 \)

Problem Statement

Find all possible rational roots of \( -4x^3 + x^2 -3x + 10 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

Brian McLogan - 4197 video solution

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

\( y = x^3 + x^2 - 10x + 8 \)

Problem Statement

Find all possible rational roots of \( y = x^3 + x^2 - 10x + 8 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

Brian McLogan - 4199 video solution

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

\( f(x) = 9x^4 - 9x^3 - 58x^2 + 4x + 24 \)

Problem Statement

Find all possible rational roots of \( f(x) = 9x^4 - 9x^3 - 58x^2 + 4x + 24 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

Brian McLogan - 4190 video solution

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

\( 2y^4 + 7y^3 - 26y^2 + 23y - 6 \)

Problem Statement

Find all possible rational roots of \( 2y^4 + 7y^3 - 26y^2 + 23y - 6 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

Brian McLogan - 4200 video solution

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

\( 3x^4 + bx^3 + ax^2 + 2 \)

Problem Statement

Find all possible rational roots of \( 3x^4 + bx^3 + ax^2 + 2 \).

Solution

Brian McLogan - 4201 video solution

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

\( f(x) = 4x^5 + x^4 - 2x^3 - 5x^2 + 8x + 16 \)

Problem Statement

Find all possible rational roots of \( f(x) = 4x^5 + x^4 - 2x^3 - 5x^2 + 8x + 16 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

Brian McLogan - 4194 video solution

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

\( x^8 - 5x^6 + x^4 + 2x - 16 \)

Problem Statement

Find all possible rational roots of \( x^8 - 5x^6 + x^4 + 2x - 16 \) and then determine which ones are actual roots. Factor the polynomial as much as possible using these roots.

Solution

Brian McLogan - 4196 video solution

video by Brian McLogan

Log in to rate this practice problem and to see it's current rating.

Really UNDERSTAND Precalculus

Log in to rate this page and to see it's current rating.

Page Resources

Topics You Need To Understand For This Page

zero product rule

functions

polynomials

zeroes/roots

To bookmark this page and practice problems, log in to your account or set up a free account.

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

how to take good notes

Try Amazon Prime 30-Day Free Trial

As an Amazon Associate I earn from qualifying purchases.

I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me.

Support 17Calculus on Patreon

Practice Search

Practice Instructions

Unless otherwise instructed, find all possible rational roots of these polynomial functions using the rational roots theorem. Determine which are actual roots and factor the polynomial as much as possible, using the roots.

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.

17calculus

Copyright © 2010-2022 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics