Fundamental Theorem of Algebra

A statement in mathematics that has been proven, or that has the potential to be proven, is referred to as a theorem.

A logical argument that uses the inference rules of a deductive system to establish that a theorem is a logical consequence of the axioms and other previously established theorems is referred to as the proof of the theorem. 

Every non-constant single-variable polynomial with complex coefficients is guaranteed to have at least one complex root, according to the fundamental theorem of algebra, also known as d’Alembert’s theorem or the d’Alembert–Gauss theorem.

This statement is also known as d’Alembert’s theorem.

 This encompasses polynomials with real coefficients due to the fact that every real number is a complex number but its imaginary part is always equal to zero.

The theorem states that the field of complex numbers is algebraically closed, which is equivalent to saying that the field is closed by definition.

The theorem can also be expressed as follows: each non-zero, single-variable, degree n polynomial with complex coefficients has precisely n complex roots when counted with multiplicity. 

By performing repeated divisions of polynomials, it is possible to demonstrate that the two assertions are equivalent to one another.

In spite of the fact that it bears the label “algebraic proof,” there is in fact no such thing. 

This is due to the fact that any proof must use some version of the analytic completeness of the real numbers, which is not a concept that is associated with algebra.

In addition to this, it is not essential for contemporary algebra.

The name for this concept was provided during a time when algebra was synonymous with the theory of equations.

Theorem of Algebra

Every nonconstant polynomial that includes complex coefficients is guaranteed to have a complex root, according to the fundamental theorem of algebra.

 In point of fact, there is not a single known demonstration of this theorem that does not involve some form of analysis. 

This is due to the fact that the outcome is dependent on particular properties of the complex numbers that are naturally best represented using topological terms.

Every complex polynomial of degree n must have n complex roots, and counting multiplicities is part of the division procedure, so this is a logical conclusion. 

To put it another way, every polynomial over C can be broken into several polynomials over C, or it can be decomposed into linear factors.

Second-Degree Equation

In mathematics, the quadratic equation is referred to as an equation of the second degree. 

The definition of a quadratic equation is the polynomial equation of the second degree with the conventional form ax² + bx+ c = 0,

 where a is not equal to zero and b and c are positive numbers. 

The answers that can be found by solving the equation are referred to as the roots of the quadratic equation. In this equation, x stands for the undetermined variable, and a, b, and c are the constants (sometimes referred to as the quadratic coefficients). If an equals zero, the equation is considered to be a linear equation.

Since there is only one variable that needs to be determined, the quadratic equation is referred to as “univariate.” The quadratic equation is a polynomial equation because it contains only powers of x that are positive integers and does not contain any negative integers. Given that two is the highest power in this equation, we may say that it is a polynomial equation of the second degree.

Conclusion

In mathematical logic, the ideas of theorems and proofs have been formalised so that mathematical reasoning may be done about them. This makes it possible to reason mathematically about them. In this setting, remarks take on the shape of whole sentences written in a more formal language. Some fundamental assertions, known as axioms, plus a set of rules for drawing conclusions make up a theory (sometimes included in the axioms). 

The theorems of the theory are statements that can be deduced from the axioms by making use of the rules for deducing things from the axioms. As a result of this formalisation, the proof theory was developed, which makes it possible to prove general theorems about theorems and proofs. 

In particular, Godel’s incompleteness theorems demonstrate that every consistent theory that includes the natural numbers also contains true statements about the natural numbers that are not theorems of the theory itself (that is they cannot be proved inside the theory).