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If we append $n$ zeros to the end of a polynomial, the polynomial does not change. |
If we append a zero to the end of a polynomial, the polynomial does not change. |
The polynomial $a + bx$ is equal to $b + ax$. |
If the first coefficient of a polynomial is zero, then the polynomial is the same as the polynomial with the first coefficient removed. |
The degree of a polynomial is less than or equal to the length of its coefficient list. |
The coefficient of a polynomial is the nth element of the list of coefficients. |
The coefficients of a polynomial are the empty list if and only if the polynomial is zero. |
If a polynomial is not zero, then its list of coefficients is not empty. |
The coefficients of the zero polynomial are the empty list. |
The coefficients of a polynomial with a leading coefficient $a$ are the same as the coefficients of the original polynomial with $a$ prepended. |
If $p$ is a nonzero polynomial, then the length of the list of coefficients of $p$ is one more than the degree of $p$. |
If $p$ is a nonzero polynomial and $n \leq \deg(p)$, then the $n$th coefficient of $p$ is equal to the $n$th coefficient of $p$. |
If $p$ is a nonzero polynomial and $n \leq \deg(p)$, then the coefficient of $x^n$ in $p$ is in the list of coefficients of $p$. |
If $p$ is a nonzero polynomial, then the list of coefficients of $p$ is the same as the list of coefficients of $p$ with $a$ prepended. |
The polynomial whose coefficients are the coefficients of $p$ is $p$. |
The coefficients of a polynomial are the same as the coefficients of the polynomial with trailing zeros removed. |
The coefficients of a polynomial are non-zero. |
The coefficients of a polynomial are never all zero. |
Two polynomials are equal if and only if their coefficients are equal. |
The $n$th coefficient of a polynomial $p$ is equal to the $n$th element of the list of coefficients of $p$. |
The coefficient of a polynomial is the nth element of the list of coefficients, or 0 if the index is out of bounds. |
If $p$ is a polynomial whose coefficients are given by the list $xs$, and $xs$ has no trailing zeros, then $p$ is equal to the polynomial whose coefficients are given by $xs$. |
The degree of a polynomial is one less than the length of its coefficient list. |
If $p$ is a nonzero polynomial, then the length of the list of coefficients of $p$ is one more than the degree of $p$. |
The list of coefficients of the zero polynomial is the empty list. |
The list of coefficients of a polynomial is the coefficient of the leading term followed by the list of coefficients of the rest of the polynomial. |
The set of coefficients of a polynomial is a subset of $\{0\}$ if and only if the polynomial is zero. |
The set of coefficients of a polynomial is never the singleton set containing only $0$. |
For all $n$, $P(coeff(p, n))$ if and only if for all $c \in coeffs(p)$, $P(c)$. |
The equality of two polynomials is always true. |
The predicate $is\_zero$ is equivalent to the predicate $p = 0$. |
The coefficients of a polynomial are the same as the list of coefficients, with leading zeros removed. |
The fold function over the coefficients of a polynomial with zero as the initial value is the identity function. |
If $f(0) = id$, then $f(a) \circ f(p) = f(a + p)$. |
The fold of a polynomial with a leading coefficient of $0$ and a constant term of $0$ is the identity function. |
If the leading coefficient of a polynomial is nonzero, then the polynomial is equal to the leading coefficient times the polynomial with the leading coefficient removed. |
If $p$ is a non-zero polynomial, then the coefficient of the leading term of $p$ is equal to the coefficient of the leading term of $p$ times the leading term of $p$. |
The polynomial $p$ is equal to the sum of the coefficients of $p$ multiplied by the corresponding powers of $x$. |
The polynomial $p(x) = 0$ is the zero polynomial. |
The polynomial $p(x) = a_0 + a_1 x + \cdots + a_n x^n$ can be written as $p(x) = a_0 + x (a_1 + x (a_2 + \cdots + x (a_{n-1} + x a_n) \cdots ))$. |
The value of a polynomial $p$ at a point $x$ is the sum of the coefficients of $p$ multiplied by $x$ raised to the corresponding power. |
The value of a polynomial at $0$ is equal to its constant coefficient. |
The coefficient of a monomial is the coefficient of the monomial if the degree of the monomial is the same as the degree of the coefficient, and zero otherwise. |
The monomial $a^0$ is equal to the polynomial $a$. |
The monomial $x^{n+1}$ is equal to $x \cdot x^n$. |
The monomial $x^n$ is equal to $0$ if $n = 0$. |
A monomial is zero if and only if its coefficient is zero. |
Two monomials are equal if and only if their coefficients are equal. |
The degree of a monomial is less than or equal to the exponent of the monomial. |
If $a \neq 0$, then the degree of the monomial $a x^n$ is $n$. |
The coefficients of a monomial are the coefficients of the constant polynomial with the same value, padded with zeros. |
If $a \neq 0$, then the fold of the coefficients of the monomial $a x^n$ is the composition of $f(0)$ with itself $n$ times, followed by $f(a)$. |
The polynomial corresponding to a monomial is the monomial itself. |
Two monomials are equal if and only if they have the same coefficient and either both are zero or both have the same degree. |
A monomial is equal to a constant polynomial if and only if the monomial is zero or the constant is zero. |
If $p$ is a nonzero polynomial, then the leading coefficient of $p$ is the same as the leading coefficient of $p$ with a new leading term. If $p$ is the zero polynomial, then the leading coefficient of $p$ is the new leading term. |
The leading coefficient of a monomial is the coefficient of the monomial. |
If $p$ is a nonzero polynomial, then the last coefficient of $p$ is equal to the leading coefficient of $p$. |
The coefficient of $x^n$ in the sum of two polynomials is the sum of the coefficients of $x^n$ in each polynomial. |
The coefficient of the $n$th term of the difference of two polynomials is the difference of the coefficients of the $n$th terms of the two polynomials. |
The coefficient of $x^n$ in $-p$ is the negative of the coefficient of $x^n$ in $p$. |
The sum of two polynomials is the polynomial whose first coefficient is the sum of the first coefficients of the two polynomials, and whose remaining coefficients are the sum of the remaining coefficients of the two polynomials. |
The negation of a polynomial with a leading coefficient $a$ is a polynomial with leading coefficient $-a$. |
The difference of two polynomials is the polynomial whose coefficients are the differences of the coefficients of the two polynomials. |
The degree of a sum of two polynomials is less than or equal to the maximum of the degrees of the two polynomials. |
If the degree of $p$ is at most $n$ and the degree of $q$ is at most $n$, then the degree of $p + q$ is at most $n$. |
If the degree of a polynomial $p$ is less than $n$, and the degree of a polynomial $q$ is less than $n$, then the degree of $p + q$ is less than $n$. |
If the degree of $p$ is less than the degree of $q$, then the degree of $p + q$ is equal to the degree of $q$. |
If the degree of $q$ is less than the degree of $p$, then the degree of $p + q$ is equal to the degree of $p$. |
The degree of a polynomial is the same as the degree of its negation. |
If the degree of $p$ is less than the degree of $q$, then the leading coefficient of $p + q$ is the same as the leading coefficient of $q$. |
The leading coefficient of a polynomial is the same as the leading coefficient of its negation. |
The degree of the difference of two polynomials is less than or equal to the maximum of the degrees of the two polynomials. |
If the degree of a polynomial $p$ is less than or equal to $n$, and the degree of a polynomial $q$ is less than or equal to $n$, then the degree of $p - q$ is less than or equal to $n$. |
If the degree of two polynomials is less than $n$, then the degree of their difference is less than $n$. |
The sum of two monomials with the same exponent is a monomial with the sum of the coefficients and the same exponent. |
The difference of two monomials is a monomial. |
$-x^n = (-x)^n$. |
The coefficient of the $i$th term of the sum of polynomials is the sum of the coefficients of the $i$th term of each polynomial. |
The monomial of the sum of a set of numbers is the sum of the monomials of the numbers. |
The coefficients of the sum of two polynomials are the sum of the coefficients of the two polynomials. |
The coefficients of $-p$ are the negations of the coefficients of $p$. |
The difference of two polynomials is the sum of the first polynomial and the negation of the second polynomial. |
The polynomial $p + q$ evaluated at $x$ is equal to the sum of the polynomials $p$ and $q$ evaluated at $x$. |
The polynomial $-p$ is the same as the polynomial $p$ with all coefficients negated. |
The difference of two polynomials is the polynomial whose value at $x$ is the difference of the values of the two polynomials at $x$. |
The polynomial of the sum of a family of polynomials is the sum of the polynomials of the family. |
If $f$ is a function from a finite set $S$ to polynomials of degree at most $n$, then the sum of the polynomials $f(s)$ for $s \in S$ has degree at most $n$. |
If $p$ is a polynomial of degree at most $n$, then $p$ is equal to the sum of its monomials of degree at most $n$. |
Every polynomial can be written as a sum of monomials. |
The polynomial whose coefficients are the elements of the list $xs$ followed by the element $x$ is equal to the polynomial whose coefficients are the elements of the list $xs$ plus the monomial $x$ times $x^n$, where $n$ is the length of the list $xs$. |
The coefficient of $x^n$ in $a p(x)$ is $a$ times the coefficient of $x^n$ in $p(x)$. |
The degree of a polynomial multiplied by a scalar is less than or equal to the degree of the polynomial. |
The product of two scalars is equal to the scalar multiple of the product of the scalars. |
For any scalar $a$, $a \cdot 0 = 0$. |
The scalar product of $0$ and any polynomial is $0$. |
Multiplying a polynomial by 1 does not change it. |
The scalar product of a polynomial with a sum of polynomials is equal to the sum of the scalar products of the polynomial with each of the summands. |
For any polynomial $p$ and any scalars $a$ and $b$, we have $(a + b)p = ap + bp$. |
For any ring $R$ and any $a \in R$, $a(-p) = -ap$. |
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