Theorem about rational zeros

The theorem about rational zeros (also the rational zero test or Gauss lemma ) is a statement about the rational zeros of integer polynomials . It contains a necessary criterion for the existence of a rational zero and provides a finite set of rational numbers in which all rational zeros must be included.

statement

For every rational zero of an integer polynomial, the numerator of its abbreviated representation divides the absolute term and the denominator divides the leading coefficient of the polynomial.

So be with a polynomial of degree and (where prime is) a rational zero of , then by divisible by divisible. ${\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{1}x+a_{0}}$ ${\displaystyle a_{i}\in \mathbb {Z} }$ ${\displaystyle n}$ ${\displaystyle x_{0}={\tfrac {p}{q}}}$ ${\displaystyle p,q\in \mathbb {Z} }$ ${\displaystyle f}$ ${\displaystyle a_{0}}$ ${\displaystyle p}$ ${\displaystyle a_{n}}$ ${\displaystyle q}$

Remarks

If the leading coefficient of the polynomial is 1, then every rational zero is an integer that divides the absolute term . ${\displaystyle a_{n}}$ ${\displaystyle a_{0}}$

The theorem can also be used to compute the rational roots of rational polynomials. Because if you multiply a rational polynomial with a common multiple of the denominators of its coefficients, you get an integer polynomial with the same zeros, for the determination of which you can now use the rational zero test.

The theorem about rational zeros also results as a corollary to a more general statement, going back to Gauss, about polynomials over the quotient field of a factorial ring (see Gauss lemma ).

Examples

The polynomial has no rational zero, since 1 and −1 are the only divisors of the absolute term and the leading coefficient and and is. ${\displaystyle p(x)=x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1}$ ${\displaystyle p(-1)=1\neq 0}$ ${\displaystyle p(1)=7\neq 0}$

Multiplying the rational polynomial by 30 gives the integer polynomial . Its rational zeros must then be included in the set . If you now check all of these candidates by inserting them into or , the zeros are 1 and . Since a polynomial of degree 3 can have a maximum of three different zeros in pairs, there are no further irrational zeros in this case. ${\displaystyle p(x)={\tfrac {1}{5}}x^{3}-{\tfrac {7}{30}}x^{2}+{\tfrac {1}{30}}}$ ${\displaystyle p^{\star }(x)=6x^{3}-7x^{2}+1}$ ${\displaystyle \left\{\pm 1,\pm {\tfrac {1}{2}},\pm {\tfrac {1}{3}},\pm {\tfrac {1}{6}}\right\}}$ ${\displaystyle p}$ ${\displaystyle p^{\star }}$ ${\displaystyle -{\tfrac {1}{3}}}$ ${\displaystyle {\tfrac {1}{2}}}$ ${\displaystyle p}$

literature

• Kurt Meyberg, Peter Vachenauer: Higher Mathematics 1 . Springer, 6th edition 2006, ISBN 3-540-41850-4 , p. 64 ( excerpt in the Google book search)
• Rolf Walter: Introduction to Analysis 1 . Walter de Gruyter 2007, ISBN 978-3-11-019539-2 , pp. 110–111, 362 ( excerpt from Google book search)
• Charles D. Miller, Margaret L. Lial, David I. Schneider: Fundamentals of College Algebra . Scott & Foresman / Little & Brown Higher Education, 3rd Revised Edition 1990, ISBN 0-673-38638-4 , pp. 216-221
• Phillip S. Jones, Jack D. Operates: The historical roots of elementary mathematics . Dover Courier Publications, 1998, ISBN 0-486-25563-8 , pp. 116–117 ( excerpt from Google book search)

Web links

• Winfried Kaballo: Analysis I . Lecture notes, winter semester 2006/2007, p. 44, chapter Polynomials and zeros , Theorem 9.11; Chapter 9 ( Memento of July 19, 2007 in the Internet Archive ) (PDF)
• The Rational Roots Test on purplemath.com
• Eric W. Weisstein : Rational Zero Theorem . In: MathWorld (English).