Univariate polynomial in the context of Word problem (mathematics education)

In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the coefficient of the nonzero term of highest degree) is equal to 1. That is to say, a monic polynomial is one that can be written as

with ${\displaystyle n\geq 0.}$

In mathematics and computer science, polynomial evaluation refers to computation of the value of a polynomial when its indeterminates are substituted for some values. In other words, evaluating the polynomial ${\displaystyle P(x_{1},x_{2})=2x_{1}x_{2}+x_{1}^{3}+4}$ at ${\displaystyle x_{1}=2,x_{2}=3}$ consists of computing ${\displaystyle P(2,3)=2\cdot 2\cdot 3+2^{3}+4=24.}$ See also Polynomial ring § Polynomial evaluation

For evaluating the univariate polynomial ${\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0},}$ the most naive method would use ${\displaystyle n}$ multiplications to compute ${\displaystyle a_{n}x^{n}}$, use ${\displaystyle n-1}$ multiplications to compute ${\displaystyle a_{n-1}x^{n-1}}$ and so on for a total of ${\displaystyle {\tfrac {n(n+1)}{2}}}$ multiplications and ${\displaystyle n}$ additions.Using better methods, such as Horner's rule, this can be reduced to ${\displaystyle n}$ multiplications and ${\displaystyle n}$ additions. If some preprocessing is allowed, even more savings are possible.

In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of degree but, nowadays, may refer to several other concepts (see Order of a polynomial (disambiguation)).

For example, the polynomial ${\displaystyle 7x^{2}y^{3}+4x-9,}$ which can also be written as ${\displaystyle 7x^{2}y^{3}+4x^{1}y^{0}-9x^{0}y^{0},}$ has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term.

In mathematics, a square-free polynomial is a univariate polynomial (over a field or an integral domain) that has no multiple root in an algebraically closed field containing its coefficients. In characteristic 0, or over a finite field, a univariate polynomial is square free if and only if it does not have as a divisor any square of a non-constant polynomial. In applications in physics and engineering, a square-free polynomial is commonly called a polynomial with no repeated roots.

The product rule implies that, if p divides f, then p divides the formal derivative f ′ of f. The converse is also true and hence, ${\displaystyle f}$ is square-free if and only if ${\displaystyle 1}$ is a greatest common divisor of the polynomial and its derivative.