Question

Integral root theorem examples

Answer 1

Answer:

Step-by-step explanation:

In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational solutions of a polynomial equation

   a n x n + a n − 1 x n − 1 + ⋯ + a 0 = 0 {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0} {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0}

with integer coefficients a i ∈ Z {\displaystyle a_{i}\in \mathbb {Z} } {\displaystyle a_{i}\in \mathbb {Z} } and a 0 , a n ≠ 0 {\displaystyle a_{0},a_{n}\neq 0} {\displaystyle a_{0},a_{n}\neq 0}. Solutions of the equation are also called roots or zeroes of the polynomial on the left side.

The theorem states that each rational solution x = p/q, written in lowest terms so that p and q are relatively prime, satisfies:

   p is an integer factor of the constant term a0, and

   q is an integer factor of the leading coefficient an.

The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials. The integral root theorem is the special case of the rational root theorem when the leading coefficient is an = 1.