Polynomial root-finding

Finding <a href="/facts/Polynomial_root/mlK3dhoT">the roots of polynomials</a> is a long-standing problem that has been extensively studied throughout the history and substantially influenced the development of mathematics. It involves determining either a numerical approximation or a closed-form expression of the roots of a univariate polynomial, i.e., determining approximate or closed form solutions of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 in the equation

 
 
 
 
 a
 
 0
 
 
 +
 
 a
 
 1
 
 
 x
 +
 
 a
 
 2
 
 
 
 x
 
 2
 
 
 +
 ⋯
 +
 
 a
 
 n
 
 
 
 x
 
 n
 
 
 =
 0
 
 
 {\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n}=0}

where 
 
 
 
 
 a
 
 i
 
 
 
 
 {\displaystyle a_{i}}
 
 are either <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>.
Efforts to understand and solve polynomial equations led to the development of important mathematical concepts, including irrational and complex numbers, as well as foundational structures in modern algebra such as fields, rings, and groups. Despite of being historically important, finding the roots of higher degree polynomials no longer play a central role in mathematics and computational mathematics, with one major exception in <a href="/facts/Computer_algebra/hjun1KAX">computer algebra</a>.

Polynomial root-finding open-in-new

Polynomial root-finding