7 Questions
What is the definition of a zero of a polynomial?
A value of the variable that makes the polynomial equal to zero.
What is the maximum number of distinct zeros that a polynomial of degree n can have?
n
What is the minimum number of complex zeros that a non-constant polynomial has?
One
What is the Rational Zero Theorem used for?
Finding all rational zeros of a polynomial with integer coefficients
What is the purpose of Descartes' Rule of Signs?
To determine the possible number of positive and negative real zeros of a polynomial
What is the result of the Conjugate Roots Theorem?
If a polynomial with real coefficients has a complex zero, then its conjugate is also a zero
What is the Linear Factorization Theorem used for?
To factor a polynomial as a product of linear factors
Study Notes
Zeroes of a Polynomial
Definition
- 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.
Properties
- 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.
Finding Zeros
- 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.
Relationships Between Zeros
- 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.
Applications
- Finding zeros of a polynomial is important in many fields, such as:
- Algebraic geometry
- Number theory
- Computer science
- Physics and engineering
Zeroes of a Polynomial
Definition
- 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.
Properties
- 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.
Finding Zeros
- 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.
Relationships Between Zeros
- 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.
Applications
- Finding zeros of a polynomial is important in many fields, such as:
- Algebraic geometry
- Number theory
- Computer science
- Physics and engineering
Learn about the definition, properties, and methods of finding the zeros of a polynomial, including the Fundamental Theorem of Algebra and the Rational Zero Theorem.
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