Rational Roots Theorem

Pronunciation: /ˈræʃ nl rutz ˈθɪər əm/ ?

The rational roots theorem gives possible rational roots of a single variable polynomial with integer coefficients. A rational roots is a root of a polynomial that is a rational number. Given a polynomial
(3x-2)(5x+7)=15x^2+11x-14,
any rational roots of the polynomial have a factor of a0 as a numerator and a factor of an as a denominator. The rational roots theorem is also called the rational zeros theorem.

Start with the polynomial 2x3 + 5x2 - 4x - 3. Since a0 = -3, the numerator of any rational roots must be one of ±1, ±3. Since a3 = 2, the denominator of any rational roots must be one of ±1, ±2.

To see why, start with the two factors
(3x-2)(5x+7).
Setting each factor to 0 gives the roots of the polynomial:
3x-2=0 implies 3x=2 implies x=3/2
and
5x+7=0 implies 5x=-7 implies x=-5/7.
The roots of the polynomial (3x+2)(5x-7) are x = 2/3 and x = -7/5.

Now multiply the two factors of the polynomial.
(3x-2)(5x+7)=3x*5x+3x*7+(-2)*5x+(-2)*7=15x^2+21x-10x-14=15x^2+11x-14.
According to the Rational Roots Theorem, any rational roots of the polynomial will be a factor of -14 divided by a factor of 15. The factors of -14 are ±1, ±2, ±7, ±14. The factors of 15 are ±1, ±3, ±5, ±15.

Examples

StepEquationDescription
1P(x)=2x^3+5x^2-4x-3?0This is the polynomial of which to find roots.
2Factors of 2 are ±1, ±2.Find all the factors of an.
3Factors of -3 are ±1, ±3.Find all the factors of a0.
4{±(1/1),±(3/1),±(1/2),±(3/2)Calculate all the possible rational roots by dividing the factors of -3 by the factors of 2.
5{±1,±3,±(1/2),±(3/2)Simplify any fractions that can be simplified.
6P(1)=2(1)^3+5(1)^2-4(1)-3?0Test the root x=1 by substituting 1 in for x.
72*1+5*1-4*1-3?0Simplify the exponents.
82+5-4-3?0Simplify the multiplication.
90?0Simplify the addition. Since 0=0 is a true statement, 1 is a root of P(x).
10(2x^3+5x^2-4x-3)/(x-1)=2x^2+7x+3Use synthetic division to find the remaining factor.
11P(x)=(x-1)(2x^2+7x+3)Here are the factors of the polynomial. Use the quadratic equation to find any roots of the quadratic 2x2+7x+3.
Example 1

StepEquationDescription
1P(x)=2x^3+5x^2-4x-3?0This is the polynomial of which to find roots.
2Factors of 2 are ±1, ±2.Find all the factors of an.
3Factors of -3 are ±1, ±3.Find all the factors of a0.
4{±(1/1),±(3/1),±(1/2),±(3/2)Calculate all the possible rational roots by dividing the factors of -3 by the factors of 2.
5{±1,±3,±(1/2),±(3/2)Simplify any fractions that can be simplified.
6P(3)=2(3)^3+5(3)^2-4(3)-3?0Test the root x=3 by substituting 3 in for x.
72*27+5*9-4*3-3?0Simplify the exponents.
854+45-21-3?0Simplify the multiplication.
984?0Simplify the addition and subtraction. Since 84≠0, 3 is not a root of P(x).
Example 2

Cite this article as:


Rational Roots Theorem. 2009-02-06. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/r/rationalrootstheorem.html.

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2009-02-06: Added vocabulary links (McAdams, David.)
2009-01-29: Initial version (McAdams, David.)

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