Polynomial Zeros

Study Notes

A zero of a polynomial is a value of the variable that makes the polynomial equal to zero.
Also known as roots or solutions of the polynomial.

Fundamental Theorem of Algebra: Every non-constant polynomial has at least one complex zero.
A polynomial of degree n has at most n distinct zeros.
If a is a zero of a polynomial, then (x - a) is a factor of the polynomial.

Rational Zero Theorem: If a rational number p/q is a zero of a polynomial with integer coefficients, then p is a factor of the constant term and q is a factor of the leading coefficient.
Synthetic Division: A method for finding zeros of a polynomial by dividing it by (x - a) and checking if the remainder is zero.
Descartes' Rule of Signs: A method for determining the possible number of positive and negative real zeros of a polynomial.

Conjugate Roots Theorem: If a polynomial with real coefficients has a complex zero, then its conjugate is also a zero.
Linear Factorization Theorem: A polynomial can be factored as a product of linear factors, each corresponding to a zero of the polynomial.

Finding zeros of a polynomial is important in many fields, such as:
- Algebraic geometry
- Number theory
- Computer science
- Physics and engineering

A value of the variable that makes the polynomial equal to zero is called a zero, also known as a root or solution of the polynomial.

The Fundamental Theorem of Algebra states that every non-constant polynomial has at least one complex zero.
A polynomial of degree n has at most n distinct zeros.
If a is a zero of a polynomial, then (x - a) is a factor of the polynomial.

The Rational Zero Theorem states that if a rational number p/q is a zero of a polynomial with integer coefficients, then p is a factor of the constant term and q is a factor of the leading coefficient.
Synthetic Division is a method for finding zeros of a polynomial by dividing it by (x - a) and checking if the remainder is zero.
Descartes' Rule of Signs is a method for determining the possible number of positive and negative real zeros of a polynomial.

The Conjugate Roots Theorem states that if a polynomial with real coefficients has a complex zero, then its conjugate is also a zero.
The Linear Factorization Theorem states that a polynomial can be factored as a product of linear factors, each corresponding to a zero of the polynomial.