## 17Calculus Precalculus - Rational Roots Test

##### 17Calculus

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

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.

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.