Polynomial arithmetic

Polynomial arithmetic is a branch of algebra dealing with some properties of polynomials which share strong analogies with properties of number theory relative to integers. It includes basic mathematical operations such as addition, subtraction, and multiplication, as well as more elaborate operations like Euclidean division, and properties related to roots of polynomials. The latter are essentially connected to the fact that the set K[X] of univariate polynomials with coefficients in a field K is a commutative ring, such as the ring of integers ${\displaystyle \mathbb {Z} }$.

Addition and subtraction of two polynomials are performed by adding or subtracting corresponding coefficients. If

then addition is defined as

Multiplication is performed much the same way as addition and subtraction, but instead by multiplying the corresponding coefficients. If ${\displaystyle f(x)=\sum _{i=0}^{n}a_{i}x^{i};g(x)=\sum _{i=0}^{m}b_{i}x^{i}}$ then multiplication is defined as ${\displaystyle f(x)\times g(x)=\sum _{i=0}^{n+m}c_{i}x^{i}}$ where ${\displaystyle c_{k}=a_{0}b_{k}+a_{1}b_{k-1}+\cdots +a_{k-1}b_{1}+a_{k}b_{0}}$. Note that we treat ${\displaystyle a_{i}}$ as zero for ${\displaystyle i>n}$ and that the degree of the product is equal to the sum of the degrees of the two polynomials.

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