One tells us how to find that integer roots of a polynomial with integer coefficients. The second tells us that all rational roots of a monic polynomial with integer coefficients are integers.

In algebra, the rational root theorem states that given an integer polynomial with leading coefficient and constant term , if has a rational root in lowest terms, then and . This theorem is most often used to guess the roots of polynomials. It sees widespread usage in introductory and intermediate mathematics competitions.

A short example shows the usage of the integer root theorem: Show that if x x is a positive rational such that x^2 + x x2 +x is an integer, then x x must be an integer.

Working our way toward the rational root theorem

The integer root theorem. If an integer is a root of a polynomial whose coefficients are integers and whose leading coefficient is ±1, then that integer is a factor of the constant term.

How to guess integer roots of polynomials in . Given polynomial with coefficients , thus any integer root of divides . As has only a finit number of divisors one can determine all integer roots of .

2024年12月21日 · To solve the problem of finding all possible integer roots of the polynomial f (x) = −x3 +4x2 −18x + 80, we can use the Rational Root Theorem. This theorem provides a way to find the possible rational roots of a polynomial equation with integer coefficients. Here’s how you can apply it step-by-step: Identify the coefficients:

Use the integer root theorem to find all the roots. Study with Quizlet and memorize flashcards containing terms like ±1, ±2, ±5, ±10, ±1, ±5, ±1, ±17 and more.

2024年5月21日 · This calculator uses the theorem to predict potential roots based on the polynomial’s coefficients, providing a list of all possible integer roots.

Theorem 1.5 (Rational Roots Theorem): Suppose f(x) = anxn+ +a0 is a polynomial with integer coe cients and with an 6= 0. Then all rational roots of f are in the form