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Fundamental theorem of algebra

fundamental theorem of algebra

The Fundamental Theorem of Algebra that every polynomial with degree n ≥ 1 has exactly n zeros where every one of them counts as many times as its multiplicity.

Polynomials are expressions consisting of variables and coefficients. The general form of a polynomial with a leading exponent n is:

a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0
Where a_n \not= 0.

Take for an example second degree polynomial. When solving quadratic equations you learned that every quadratic polynomial has exactly two zeros or roots. One way of solving these equations is factorization. If you remember, if you could represent your polynomial as a product of two linear polynomials you could easily see the zeros. This is why this theorem can be said in another way.

Every polynomial with degree  can be, in a unique way, presented as the product of n linear polynomials.

The second form of this theorem is:

Example 1. Find all zeros of the function \f(x) = x^2 + 2x + 1.

Function \f(x) = x^2 + 2x + 1 can be written as \ f(x) = (x + 1)^2 which means that 1 is the zero of this function. But, according to the Fundamental Theorem of Algebra, this polynomial has two zeros. This means that 1 is the zero with multiplicity 2.

If we had function \ f(x) = (x + 2)^2 (x + 3) its zeros would be 3 and – 2 with multiplicity 2.

Of course, these forms of polynomials are much easier to solve. This is why we always tend to factor polynomials.