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The $n$th derivative of $n$ is zero. |
The coefficient of the $n$th term of the derivative of a polynomial $p$ is $n+1$ times the coefficient of the $(n+1)$st term of $p$. |
The $n$th coefficient of the derivative of a polynomial is equal to the $n$th coefficient of the polynomial multiplied by $n$. |
The coefficients of the derivative of a polynomial are given by the function pderiv_coeffs. |
The derivative of a polynomial is zero if and only if the polynomial is constant. |
The degree of the derivative of a polynomial is one less than the degree of the polynomial. |
If $p$ is a non-constant polynomial, then $p$ does not divide its derivative. |
A polynomial $p$ divides its derivative $p'$ if and only if $p$ is a constant polynomial. |
The derivative of the constant polynomial $a$ is the zero polynomial. |
The derivative of the sum of two polynomials is the sum of their derivatives. |
The derivative of the negation of a polynomial is the negation of the derivative of the polynomial. |
The derivative of the difference of two polynomials is the difference of their derivatives. |
The derivative of a polynomial multiplied by a constant is the derivative of the polynomial multiplied by the same constant. |
The derivative of the product of two polynomials is the sum of the product of the first polynomial with the derivative of the second polynomial and the product of the second polynomial with the derivative of the first polynomial. |
The $n$th derivative of $p^n$ is $n p^{n-1} p'$. |
The derivative of the composition of two polynomials is the composition of the derivatives of the two polynomials, multiplied by the derivative of the second polynomial. |
The derivative of a product of functions is the sum of the products of the derivatives of the functions. |
The derivative of $x^{n+1}$ is $(n+1)x^n$. |
If $f$ is differentiable at $x$, then $a + f$ is differentiable at $x$. |
The derivative of a polynomial is the polynomial whose coefficients are the derivatives of the coefficients of the original polynomial. |
The function $x \mapsto p(x)$ is continuous at $x$. |
If $f$ tends to $a$ in $F$, then the polynomial $p(f(x))$ tends to $p(a)$ in $F$. |
The function $p(x)$ is continuous at $z$ on the set $s$. |
If $f$ is a continuous function from a topological space $F$ to a real normed field, then the function $x \mapsto p(f(x))$ is continuous. |
If $f$ is a continuous function from $A$ to $\mathbb{R}$, then the function $x \mapsto p(f(x))$ is continuous on $A$. |
The function $x \mapsto p(x)$ is differentiable at $x$ for any real number $x$. |
If $a < b$ and $p(a) < 0 < p(b)$, then there exists $x$ such that $a < x < b$ and $p(x) = 0$. |
If $a < b$, $p(a) > 0$, and $p(b) < 0$, then there exists $x$ such that $a < x < b$ and $p(x) = 0$. |
If $p$ is a polynomial with real coefficients and $p(a)p(b) < 0$, then there exists a root of $p$ in the interval $(a,b)$. |
If $a < b$, then there exists $x$ such that $a < x < b$ and $p(b) - p(a) = (b - a)p'(x)$. |
If $A$ is an interval containing $[a,b]$, then there exists $x \in A$ such that $p(b) - p(a) = (b-a)p'(x)$. |
If the leading coefficient of a polynomial is positive, then there exists a number $n$ such that for all $x \geq n$, the value of the polynomial at $x$ is greater than or equal to the leading coefficient. |
The derivative of $f(x) = (x - a)^n q(x)$ is $f'(x) = (x - a)^n q'(x) + n (x - a)^{n - 1} q(x)$. |
If $p$ is a polynomial of degree $n > 0$ such that $p' \neq 0$ and $p = (x - a)^n q$ where $q$ is not divisible by $x - a$, then $n = 1 + \text{order}_a(p')$. |
If $p$ is a nonzero polynomial and $a$ is a root of $p$, then the order of $a$ as a root of $p$ is one more than the order of $a$ as a root of $p'$. |
If $p$ is a polynomial with nonzero derivative, and $p$ can be written as $qd$ where $d$ is a squarefree polynomial, then the order of $a$ in $q$ is either $0$ or $1$. |
If $p$ is a polynomial with nonzero derivative, and $p = qd$ where $p' = ed$, and $d = rp + sp'$, then $q$ is squarefree. |
If $p$ is a nonzero polynomial and $a$ is a root of $p$, then $a$ is a root of $p'$ if and only if $a$ is a root of $p$ of multiplicity $n + 1$. |
If the derivative of a polynomial is zero, then the polynomial is a constant. |
A polynomial $p$ is square-free if and only if it has no repeated roots. |
If $p$ is a polynomial with nonzero derivative, and $p = qd$ where $d$ is the squarefree part of $p$, then $q$ is squarefree and $p$ and $q$ have the same roots. |
If $p$ is a polynomial with integer coefficients, $p$ is not the zero polynomial, and $p(x) = 0$, then $x$ is algebraic. |
If $x$ is an algebraic number, then there exists a polynomial $p$ with integer coefficients such that $p(x) = 0$. |
An element $x$ is algebraic if and only if there exists a polynomial $p$ with rational coefficients such that $p(x) = 0$ and $p \neq 0$. |
An element $x$ of a field is an algebraic integer if and only if there exists a monic polynomial $p$ with integer coefficients such that $p(x) = 0$. |
If $x$ is an algebraic integer and $x \in \mathbb{Q}$, then $x \in \mathbb{Z}$. |
If $x$ is an algebraic integer, then $x$ is an algebraic number. |
Every integer is an algebraic integer. |
The integers $0$, $1$, $n$, $k$, and $m$ are algebraic integers. |
$\i$ is an algebraic integer. |
If $x$ is an algebraic integer, then so is $-x$. |
If $x$ is an algebraic integer, then $-x$ is an algebraic integer. |
If $p$ is a polynomial with integer coefficients such that $p(x) = 0$ and $p(0) = 1$, then $1/x$ is an algebraic integer. |
If $y$ is an algebraic integer and $p(x)$ is a polynomial with integer coefficients such that $p(x) = y$ and $p$ has positive degree, then $x$ is an algebraic integer. |
If $x$ is an algebraic integer, then so is $\lvert x \rvert$. |
If $x$ is an algebraic integer, then $x^{1/n}$ is an algebraic integer. |
If $x$ is an algebraic integer, then $\sqrt{x}$ is an algebraic integer. |
If $x$ is an algebraic integer, then $\sqrt{x}$ is an algebraic integer. |
The complex conjugate of a polynomial evaluated at a complex conjugate is the complex conjugate of the polynomial evaluated at the original complex number. |
If $x$ is an algebraic integer, then $\overline{x}$ is an algebraic integer. |
The conjugate of an algebraic integer is an algebraic integer. |
If $x$ is an algebraic integer, then $\text{of_real}(x)$ is an algebraic integer. |
The real number $x$ is algebraic if and only if the complex number $x$ is algebraic. |
If $d$ is a nonzero polynomial, $lc$ is the leading coefficient of $d$, $r$ is a polynomial, $dr$ is the degree of $d * r$, $n$ is a natural number, and $q'$ is the result of calling divide_poly_main with these arguments, then $q' = q + r$. |
If $r = 0$, then $divide\_poly\_main(0, 0, r, d, dr, n) = 0$. |
If $g$ is a nonzero polynomial, then $(f \cdot g) / g = f$. |
The quotient of a polynomial by the zero polynomial is the zero polynomial. |
If $c$ divides $p$, then $p/c$ is the polynomial obtained by dividing each coefficient of $p$ by $c$. |
If $a \neq 0$, then $a$ is a unit in the polynomial ring $\mathbb{Z}[x]$. |
If $a$ is a nonzero element of a field, then $a$ is a unit. |
A polynomial $p$ is a unit if and only if its degree is zero. |
A polynomial $p$ is a unit if and only if $p$ is a constant polynomial and $p$ is not zero. |
If $p$ is a unit, then $p$ is a monomial. |
A constant polynomial is a unit if and only if the constant is a unit. |
If $p$ divides $1$, then $p$ is a constant polynomial and the constant divides $1$. |
If $p$ is a unit polynomial, then $p$ is a monomial of the form $ax^0$ for some $a \neq 0$. |
A polynomial $p$ is a unit if and only if it is a constant polynomial and the constant is a unit. |
If $d$ is a nonzero polynomial, $lc$ is the leading coefficient of $d$, $r$ is a polynomial with degree at most $dr$, $q$ is a polynomial, and $n$ is a natural number, then the result of pseudo_divmod_main is a pair $(q',r')$ such that $r' = 0$ or $r'$ has degree less than the degree of $d$, and $lc^n(d q + r) = d q' +... |
If $g \neq 0$ and $f = gq + r$, then $g$ divides $f$ and $r = 0$ or $r$ has degree less than $g$. |
The second component of the result of pseudo_divmod_main is the same for two different inputs. |
If $g$ is a nonzero polynomial, then there exist a nonzero constant $a$ and a polynomial $q$ such that $a f = g q + r$, where $r$ is the pseudo remainder of $f$ and $g$. Moreover, $r$ is either zero or has degree less than the degree of $g$. |
If $d$ is a nonzero polynomial, then the first component of the result of pseudo_divmod_main is equal to the result of divide_poly_main. |
If $g$ is a nonzero polynomial and $f = gq + r$ is the pseudo-division of $f$ by $g$, then $f = gq + r$. |
If $d$ is a nonzero polynomial, then the result of dividing $d$ by $q$ is the same as the result of dividing $d$ by $q$ after scaling $q$ by a power of $lc(d)$. |
The quotient of two polynomials is equal to the first component of the pseudo-division of the first polynomial by the second. |
The normalization of a polynomial is the same as the polynomial with each coefficient divided by the unit factor of the leading coefficient. |
The coefficient of the normalized polynomial is the coefficient of the original polynomial divided by the unit factor of the leading coefficient of the original polynomial. |
The unit factor of a polynomial with a leading coefficient is the unit factor of the leading coefficient if the polynomial is constant, and the unit factor of the rest of the polynomial otherwise. |
The normalization of a monomial is the monomial with the same exponent and the normalized coefficient. |
The unit factor of a monomial is the unit factor of its coefficient. |
The normalization of a constant polynomial is the constant polynomial with the normalized constant. |
If $p$ is a polynomial and $c$ is a scalar, then the normalization of $cp$ is equal to the normalization of $c$ times the normalization of $p$. |
The Euclidean relation holds for two polynomials $x$ and $y$ if and only if $x = qy + r$ and $r = 0$ or $deg(r) < deg(y)$. |
The Euclidean relation between $0$ and $y$ is $(0, 0)$. |
The Euclidean relation between $x$ and $0$ is $(0, x)$. |
If $x$ and $y$ are polynomials with $y \neq 0$ and $x = qy + r$ for some polynomials $q$ and $r$, then $ax + by = aqy + ar + by = (a + bq)y + ar$ for any $a$ and $b$. |
There exists a polynomial $q$ and a remainder $r$ such that $x = qy + r$. |
If $(q_1, r_1)$ and $(q_2, r_2)$ are both pairs of polynomials that satisfy the Euclidean relation for polynomials, then $q_1 = q_2$ and $r_1 = r_2$. |
The Euclidean relation for polynomials is true if and only if the quotient and remainder are both zero. |
The Euclidean relation for polynomials is true if and only if the quotient is zero and the remainder is the original polynomial. |
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