Roots of polynomials

Polynomial roots, or zeros, are the values for which the polynomial equates to zero. These outcomes are obtained by setting the polynomial equal to zero and solving for x.
The degree of a polynomial indicates the highest power of x in it. It defines the maximum number of real roots the polynomial can have.
Complex roots of polynomials often come as conjugate pairs. This is because if a + bi is a root, its conjugate a - bi will also be a root for the polynomial.
The Fundamental Theorem of Algebra states that a polynomial of degree n will have exactly n roots, including multiple and complex roots.
The Rational Root Theorem can help to identify potential rational roots of a polynomial. This theorem suggests that any rational root, written in its simplest form, p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient.
The Descartes’ Rule of Signs can be used to determine the number of positive and negative roots. The number of sign changes in the polynomial provides an upper bound to the number of positive real roots.
The Remainder Theorem and Factor Theorem apply to polynomials. The Remainder Theorem states that for any polynomial f(x) and any number a, the remainder on division by (x - a) is f(a). In contrast, the Factor Theorem posits that a polynomial f(x) has a factor (x - a) if and only if f(a) = 0.
Synthetic division is a valuable method to divide a polynomial by a linear binomial and find its roots more rapidly.
Graphs of polynomials can assist in visualizing the behavior of the polynomial. The polynomial’s degree, sign of the leading coefficient, and roots will impact the shape of the curve.