Can anyone give an intuitive proof of the theorem on polynomial factorization?

In summary, the conversation was about an algebraic theorem on polynomials, which states that any polynomial can be written as a product of linear factors and irreducible quadratic factors. The person was looking for an intuitive proof of this theorem and asked for help on a math forum, where they received an answer but did not fully understand it. The theorem applies to polynomials with real coefficients and involves finding the roots, which can be expressed as a product of linear terms and combined to form irreducible quadratic terms with real constants.
  • #1
Shahed al mamun
5
0
Recently I have known an algebraic theorem on polynomial while learning the method of partial fraction.
The theorem is :
Any polynomial can be written as the product of linear factors and irreducible quadratic factors.
I did not find an intuitive proof of this theorem.I asked a question in this link http://math.stackexchange.com/questions/1158560/need-of-an-intuitive-proof-of-an-algebraic theorem

and an answer was given there but I did not fully understand this.Can anyone prove the theorem intuitively?
 
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  • #2
The theorem you stated applies to polynomials with real coefficients and the factoring terms all involve real numbers. The fundamental theorem of algebra states that the polynomial can be expressed as a product of linear terms involving the roots (times the lead coefficient). Since the coefficients are all real, complex roots will appears as conjugate pairs. Combining a pair gives an irreducible quadratic term with real constants.
 

1. What is "Proof of an algebraic theorem"?

"Proof of an algebraic theorem" refers to the process of providing logical and mathematical evidence to support the validity of a given algebraic statement or equation.

2. Why is it important to prove algebraic theorems?

Proving algebraic theorems allows for a deeper understanding and application of mathematical concepts. It also ensures that the statements and equations used in algebraic calculations are accurate and reliable.

3. What are the steps involved in proving an algebraic theorem?

The steps involved in proving an algebraic theorem may vary depending on the specific theorem, but generally involve clearly stating the theorem, providing logical reasoning and mathematical evidence, and concluding with a clear and concise summary.

4. What are some common techniques used in proving algebraic theorems?

Some common techniques used in proving algebraic theorems include mathematical induction, proof by contradiction, direct proof, and proof by exhaustion. Other techniques may also be used depending on the complexity of the theorem.

5. How can I improve my ability to prove algebraic theorems?

Improving your ability to prove algebraic theorems requires practice and a solid understanding of mathematical concepts. It can also be helpful to study and analyze proofs from other mathematicians, and to seek guidance from a teacher or mentor.

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