Rational Root Theorem and Factor Theorem

The Rational Root Theorem states that any rational solution (or root) of a polynomial equation, in the form of p/q, is such that p is a factor of the constant term and q is a factor of the leading coefficient.

To find all possible rational roots of a polynomial, identify the factors of the constant term (p) and the leading coefficient (q), then form all possible combinations of p/q.

±1, ±3, ±9, ±1/2, ±3/2, ±9/2.

1/3, 2/3, 1, 2, 3, 6, -1/3, -2/3, -1, -2, -3, -6.

To determine if a number is a root of a polynomial, substitute the number into the polynomial equation. If the result equals zero, then it is a root.

The Fundamental Theorem of Algebra states that every non-constant polynomial equation of degree n has exactly n roots in the complex number system.

A polynomial of degree 4 has exactly 4 roots, counting multiplicities.

The zeros are -1, 4, 5, 3/2.

A zero of a polynomial is a value of x that makes the polynomial equal to zero.

If x = r is a zero of a polynomial, then (x - r) is a factor of that polynomial.