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If $p$ and $q$ are positive polynomials, then $pq$ is a positive polynomial. |
A polynomial is either zero, or positive, or negative. |
A polynomial is positive if and only if its coefficients are non-empty and the last coefficient is positive. |
The synthetic division of $0$ by $c$ is $(0, 0)$. |
The synthetic division of a polynomial $p$ by a constant $c$ is the same as the synthetic division of $p$ by $c$ followed by the synthetic division of the remainder by $c$. |
The synthetic division of a polynomial by $0$ is $0$. |
If $p$ is a polynomial and $c p = a p + q$, then $p = 0$. |
The remainder of the synthetic division of a polynomial $p$ by a constant $c$ is equal to the remainder of the division of $p$ by $c$. |
The synthetic division of a polynomial $p$ by a constant $c$ is equal to the synthetic division of the polynomial $p$ by the constant $c$ plus the constant $p(c)$. |
The synthetic division of a polynomial $p$ by a constant $c$ is zero if and only if $p$ is a constant polynomial. |
The degree of the synthetic division of a polynomial $p$ by a constant $c$ is one less than the degree of $p$. |
If $p$ is a polynomial and $c$ is a constant, then $p + c \cdot \text{syntheticDiv}(p, c) = \text{pCons}(p, c) \cdot \text{syntheticDiv}(p, c)$. |
If $p + cq = rq + c$, then $r = p(c)$ and $q = \text{synthetic-div}(p, c)$. |
The synthetic division algorithm is correct. |
A polynomial $p$ is equal to zero if and only if $x - c$ divides $p$. |
For any polynomial $p$ and any number $c$, $p$ is divisible by $x - c$ if and only if $p(c) = 0$. |
If $p$ is a nonzero polynomial, then the set of roots of $p$ is finite. |
Two polynomials are equal if and only if their coefficients are equal. |
A polynomial $p$ is the zero polynomial if and only if $p(x) = 0$ for all $x$. |
The coefficient of $x^n$ in the polynomial $[a, 1]^n$ is $1$. |
The degree of the $n$th power of a linear polynomial is $n$. |
If $p$ is a polynomial with $p(a) = 0$, then $p$ is divisible by $(x - a)$ to the power of the order of $a$ in $p$. |
If $p$ is a nonzero polynomial and $a$ is a root of $p$, then $p$ is not divisible by $(x - a)^{n + 1}$ for any $n$. |
If $p$ is a nonzero polynomial and $a$ is a root of $p$, then $p$ is divisible by $(x - a)^k$ but not by $(x - a)^{k + 1}$, where $k$ is the multiplicity of $a$ as a root of $p$. |
If $p$ is a nonzero polynomial, then the order of $a$ modulo $p$ is at most the degree of $p$. |
A polynomial $p$ has a root $a$ if and only if $p = 0$ or the order of $a$ as a root of $p$ is nonzero. |
If $p(a) \neq 0$, then the order of $a$ in $p$ is $0$. |
If $p$ is a polynomial with integer coefficients and $a$ is an integer, then the order of $a$ in $p$ is the largest integer $n$ such that $x^n$ divides $p$. |
If $p$ and $q$ are nonzero polynomials, then the order of $a$ in $pq$ is the sum of the orders of $a$ in $p$ and $q$. |
If $c$ is a nonzero constant and $p$ is a polynomial, then the order of $x$ in $cp$ is the same as the order of $x$ in $p$. |
The order of $x$ modulo 1 is 0. |
The order of $x$ in $-p$ is the same as the order of $x$ in $p$. |
The order of $a$ in the polynomial ring $F[x]/(x^n - a)$ is $n$. |
If $c \neq 0$, then the order of the monomial $cx^n$ is $n$. |
If $p$ divides $q$ and $q \neq 0$, then the order of $p$ is less than or equal to the order of $q$. |
The $n$th power of the minimal polynomial of $a$ divides $p$ if and only if $p = 0$ or $n \leq$ the order of $a$ modulo $p$. |
If $p$ is a nonzero polynomial, then there exists a polynomial $q$ such that $p = (x - a)^{ord_a(p)} q$ and $(x - a)$ does not divide $q$. |
If $p$ is a nonzero polynomial, then $x^n$ divides $p$ if and only if $n \leq \text{order}(0, p)$. |
If $Q(0)$ holds and if $Q(p)$ holds whenever $P(a)$ holds and $p$ has no roots, then $Q(p)$ holds. |
If we drop all the leading occurrences of $a$ from a list of $n$ $a$'s followed by a list $ys$, we get the same result as if we dropped all the leading occurrences of $a$ from $ys$. |
If you append $n$ zeros to the end of a polynomial, the result is the same as the original polynomial. |
If $P(0,0)$ holds and $P(p,q)$ implies $P(p',q')$ for all $p'$ and $q'$ that can be obtained from $p$ and $q$ by adding a new leading coefficient, then $P(p,q)$ holds for all polynomials $p$ and $q$. |
The composition of the zero polynomial with any polynomial is the zero polynomial. |
The composition of a polynomial with a constant polynomial is equal to the sum of the constant polynomial and the product of the constant polynomial and the composition of the original polynomial with the constant polynomial. |
The polynomial $p(1)$ is equal to $1$. |
The polynomial $p \circ q$ is the same as the polynomial $p$ evaluated at the polynomial $q$. |
The degree of the composition of two polynomials is less than or equal to the product of their degrees. |
The composition of a polynomial with the sum of two polynomials is equal to the sum of the compositions of the polynomial with each of the two polynomials. |
If $p$ and $r$ are polynomials, then $-p \circ r = -(p \circ r)$. |
The composition of a polynomial with the difference of two other polynomials is the difference of the compositions of the polynomials with the third polynomial. |
If $p$ and $r$ are polynomials, then $(ap) \circ r = a(p \circ r)$. |
If $p$ and $q$ are polynomials, then $p \circ (q \circ r) = (p \circ q) \circ r$. |
The composition of polynomials is associative. |
The polynomial $p$ composed with the identity function on the interval $[0, 1]$ is equal to $p$. |
The composition of a sum of polynomials is the sum of the compositions of the polynomials. |
The composition of a product of polynomials with another polynomial is the product of the compositions of each polynomial with the other polynomial. |
The polynomial $p(x) = a$ composed with any other polynomial $q(x)$ is $p(x) = a$. |
The composition of a polynomial $p$ with the zero polynomial is the constant polynomial with value $p(0)$. |
The degree of the composition of two polynomials is the product of their degrees. |
If the composition of two polynomials is zero, and the second polynomial has degree greater than zero, then the first polynomial is zero. |
If $q$ is a non-constant polynomial, then the leading coefficient of the composition of $p$ and $q$ is the product of the leading coefficient of $p$ and the leading coefficient of $q$ raised to the degree of $p$. |
If the coefficients of $p$ and $q$ are in $R$, then the coefficients of $pq$ are in $R$. |
If the coefficients of two polynomials are in a semiring, then the coefficients of their composition are also in the semiring. |
If we drop the first $n$ elements of a list $xs$, then the $m$th element of the resulting list is the $(m+n)$th element of $xs$. |
If $n$ is the length of a list $xs$, then the $m$th element of $xs$ is the same as the $m$th element of $xs$ truncated to length $n$. |
The coefficient of $x^i$ in the polynomial $p(x + n)$ is the coefficient of $x^{i + n}$ in $p(x)$. |
The polynomial $x \mapsto x$ is the same as the polynomial $x \mapsto x + 0$. |
The polynomial $p(x)$ shifted by $n$ is $0$ if $p(x)$ is $0$. |
The polynomial $p(x) = x^n$ shifted by 1 is the polynomial $p(x-1)$. |
If $p$ is a polynomial and $n$ is a nonnegative integer, then $p(x + n)$ is the polynomial obtained by shifting the coefficients of $p$ by $n$ places to the right. |
The coefficients of a polynomial shifted by $n$ are the coefficients of the original polynomial with the first $n$ coefficients dropped. |
The coefficient of $x^k$ in the polynomial $p$ truncated to degree $n$ is the coefficient of $x^k$ in $p$ if $k < n$, and $0$ otherwise. |
The polynomial $p_n(x)$ is zero at $x = 0$. |
The polynomial cutoff of $1$ is $1$ if $n \neq 0$ and $0$ if $n = 0$. |
The coefficients of the polynomial obtained by truncating the coefficients of $p$ to the first $n$ terms are the first $n$ coefficients of $p$ with trailing zeros removed. |
The coefficients of the polynomial $p$ are the same as the coefficients of the polynomial $p$ with the leading zeros removed. |
The reflection of the polynomial $0$ is $0$. |
The reflection of the polynomial $1$ is $1$. |
The coefficient of $x^n$ in the polynomial $p(x)$ is equal to the coefficient of $x^{degree(p) - n}$ in the polynomial $reflect_poly(p)$. |
The coefficient of the constant term of the reflected polynomial is zero if and only if the original polynomial is zero. |
The polynomial $p$ is zero if and only if the reflected polynomial $p^*$ is zero at $0$. |
If $p$ is a nonzero polynomial, then the reflection of $p$ is equal to the reflection of $p$ with a monomial added to the front. |
The reflection of a constant polynomial is itself. |
If $x \neq 0$, then the polynomial $p$ reflected about the $y$-axis is equal to $x^{n}p(1/x)$, where $n$ is the degree of $p$. |
The coefficient of the leading term of a polynomial is equal to the coefficient of the constant term of the reflected polynomial. |
The leading coefficient of a polynomial is equal to the value of the polynomial at $0$. |
If the constant coefficient of a polynomial $p$ is nonzero, then the polynomial obtained by reflecting $p$ about the $y$-axis is $p$ itself. |
The degree of the reflected polynomial is less than or equal to the degree of the original polynomial. |
If $a \neq 0$, then the polynomial $a + p$ is equal to the polynomial whose coefficients are the coefficients of $p$ in reverse order, with $a$ prepended. |
If the constant coefficient of a polynomial $p$ is nonzero, then the degree of $p$ is equal to the degree of $p$ reflected about the $y$-axis. |
The reflection of a product of polynomials is the product of the reflections of the polynomials. |
The reflection of a polynomial multiplied by a constant is the reflection of the polynomial multiplied by the same constant. |
The reflection of a polynomial to the power $n$ is the reflection of the polynomial to the power $n$. |
The reflection of a product of polynomials is the product of the reflections of the polynomials. |
The product of a list of polynomials is equal to the product of the list of the reflected polynomials. |
If the list $xs$ does not end with $0$, then the polynomial $p$ with coefficients $xs$ is equal to the polynomial with coefficients $xs$ in reverse order. |
Simplification rules for polynomials. |
The derivative of the zero polynomial is the zero polynomial. |
The derivative of a polynomial $a_0 + a_1 x + \cdots + a_n x^n$ is $a_1 + 2 a_2 x + \cdots + n a_n x^{n-1}$. |
The derivative of the constant polynomial $1$ is $0$. |
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