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.
video by PatrickJMT |
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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
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
video by The Organic Chemistry Tutor |
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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
video by The Organic Chemistry Tutor |
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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
video by The Organic Chemistry Tutor |
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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
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
video by Mario's Math Tutoring |
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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
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
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
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
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
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
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
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
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
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
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
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
video by Brian McLogan |
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