






Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Learn about Gauss' theorem for factoring polynomials and see examples of factoring quadratic and cubic polynomials. Understand the concept of completely factored polynomials and how to find their roots.
What you will learn
Typology: Study notes
1 / 11
This page cannot be seen from the preview
Don't miss anything!
Any natural number that is greater than 1 can be factored into a product of prime numbers. For example 20 = (2)(2)(5) and 30 = (2)(3)(5).
In this chapter we’ll learn an analogous way to factor polynomials.
A monic polynomial is a polynomial whose leading coe cient equals 1. So x 4 2 x 3 + 5x 7 is monic, and x 2 is monic, but 3x 2 4 is not monic.
The following result tells us how to factor polynomials. It essentially tells us what the “prime polynomials” are:
Any polynomial is the product of a real number, and a collection of monic quadratic polynomials that do not have roots, and of monic linear polynomials.
This result is called the Fundamental Theorem of Algebra. It is one of the most important results in all of mathematics, though from the form it’s written in above, it’s probably di cult to immediately understand its importance. The explanation for why this theorem is true is somewhat di cult, and it is beyond the scope of this course. We’ll have to accept it on faith.
Examples.
Any natural number that is greater than 1 can be factored into a product of prime numbers. For example 20 = (2)(2)(5) and 30 = (2)(3)(5).
In this chapter we’ll learn an analogous way to factor polynomials.
A monic polynomial is a polynomial whose leading coe cient equals 1. So x^4 2 x^3 + 5x 7 is monic, and x 2 is monic, but 3x^2 4 is not monic.
Carl Friedrich Gauss was the boy who discovered a really quick way to see that 1 + 2 + 3 + · · · + 100 = 5050. In 1799, a grown-up Gauss proved the following theorem:
Any polynomial is the product of a real number, and a collection of monic quadratic polynomials that do not have roots, and of monic linear polynomials.
This result is called the Fundamental Theorem of Algebra. It is one of the most important results in all of mathematics, though from the form it’s written in above, it’s probably di cult to immediately understand its importance. The explanation for why this theorem is true is somewhat di cult, and it is beyond the scope of this course. We’ll have to accept it on faith.
Examples.
/\ /
Lf. .5 2. IS /\ /
22 35
Fundamental Theorem of Algebra