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17Calculus Precalculus - Rational Roots Test

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Rational Roots Test

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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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.

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