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For any ring $R$ and any $a \in R$, we have $-a \cdot p = -(a \cdot p)$. |
For any ring $R$ and any $a \in R$, we have $a(p - q) = ap - aq$. |
For any two scalars $a$ and $b$, and any polynomial $p$, we have $(a - b)p = ap - bp$. |
The distributive laws for scalar multiplication. |
For any scalar $a$ and any polynomial $p$, $a \cdot (b + p) = a \cdot b + a \cdot p$. |
The product of a scalar and a monomial is a monomial. |
The scalar multiple of a polynomial is the polynomial whose coefficients are the scalar multiples of the original coefficients. |
The degree of a scalar multiple of a polynomial is the same as the degree of the polynomial, unless the scalar is zero. |
If $p$ is a polynomial and $a$ is a scalar, then $ap = 0$ if and only if $a = 0$ or $p = 0$. |
The coefficients of a polynomial multiplied by a scalar are the coefficients of the original polynomial multiplied by the scalar. |
If $b \neq 0$, then $a p = b q$ if and only if $(a / b) p = q$. |
The product of a polynomial and the zero polynomial is the zero polynomial. |
If $p$ and $q$ are polynomials, then $ap + q = aq + p$. |
The product of a polynomial and the zero polynomial is the zero polynomial. |
For any polynomial $p$ and any number $a$, we have $p(x) \cdot (a + xq(x)) = ap(x) + x(p(x) \cdot q(x))$. |
The product of a polynomial and zero is zero. |
The product of a scalar and a polynomial is equal to the scalar times the product of the polynomial and the scalar. |
The product of a polynomial and a scalar multiple of another polynomial is equal to the scalar multiple of the product of the two polynomials. |
For any polynomials $p$, $q$, and $r$, $(p + q)r = pr + qr$. |
The coefficient of the highest degree term of the product of two polynomials is the product of the coefficients of the highest degree terms of the two polynomials. |
The coefficient of $x^n$ in the product of two polynomials $p$ and $q$ is the sum of the products of the coefficients of $p$ and $q$. |
The degree of a product of two polynomials is less than or equal to the sum of the degrees of the two polynomials. |
The product of two monomials is a monomial. |
The coefficient of $x^n$ in the polynomial $1$ is $1$ if $n = 0$ and $0$ otherwise. |
$1 = [1]$. |
The polynomial $[1]$ is equal to $1$. |
The polynomial $p(x) = 1$ is equal to the constant function $1$. |
The polynomial $1$ is equal to the constant polynomial $1$. |
The degree of the polynomial $1$ is $0$. |
The coefficients of the polynomial $1$ are $[1]$. |
The scalar multiple of $1$ by $c$ is the constant polynomial $c$. |
The monomial $x^0$ is equal to $1$. |
A monomial is equal to 1 if and only if its coefficient is 1 and its degree is 0. |
The monomial $cx^n$ is equal to $c[x, 1]^n$. |
The degree of a polynomial raised to a power is less than or equal to the degree of the polynomial times the power. |
The coefficient of the constant term of a polynomial $p$ raised to the $n$th power is the $n$th power of the coefficient of the constant term of $p$. |
The polynomial $a p$ is equal to $a$ times the polynomial $p$. |
The value of a polynomial at a point is equal to the product of the values of its factors at that point. |
The $n$th power of a polynomial is the polynomial whose value at $x$ is the $n$th power of the value of the original polynomial at $x$. |
The product of polynomials is a polynomial. |
If $S$ is a finite set, then the degree of the product of the polynomials in $S$ is less than or equal to the sum of the degrees of the polynomials in $S$. |
The coefficient of $x^0$ in the product of a list of polynomials is the product of the coefficients of $x^0$ in each polynomial. |
The coefficient of $x^k$ in the product of a monomial $cx^n$ and a polynomial $p$ is $0$ if $k < n$, and $c$ times the coefficient of $x^{k-n}$ in $p$ otherwise. |
A monomial $x^n$ divides a polynomial $p$ if and only if all the coefficients of $p$ up to degree $n-1$ are zero. |
The polynomial $f(0)$ is the zero polynomial. |
The polynomial $f(x)$ evaluated at $1$ is equal to $f(1)$. |
If $f(1) = 1$, then $f(x) = 1$ for all $x$. |
If $f(0) = 0$, then the coefficient of $x^n$ in $f(p(x))$ is $f(a_n)$, where $a_n$ is the coefficient of $x^n$ in $p(x)$. |
The coefficients of the polynomial obtained by applying a function to the coefficients of a polynomial are the coefficients of the original polynomial with zeros stripped off and the function applied to each coefficient. |
If $f$ is a function such that $f(x) \neq 0$ for all $x \neq 0$, then the coefficients of the polynomial $f(p)$ are the images of the coefficients of $p$ under $f$. |
If $f$ is a function such that $f(x) = 0$ if and only if $x = 0$, then the coefficients of $f(p)$ are exactly the images of the coefficients of $p$ under $f$. |
If $f$ is a function such that $f(x) \neq 0$ for all $x \neq 0$, then the degree of $f(p)$ is the same as the degree of $p$. |
If $f(0) = 0$ and $f(x) \neq 0$ for all nonzero coefficients of $p$, then $f(p) = 0$ if and only if $p = 0$. |
If $f$ is a function satisfying $f(0) = 0$ and $f(cx) = f(c)f(x)$ for all $c, x \in \mathbb{C}$, then $f(cx^n) = f(c)f(x)^n$ for all $c, x \in \mathbb{C}$ and $n \in \mathbb{N}$. |
If $f(0) = 0$, then $f(cx^n + p(x)) = f(c)x^n + f(p(x))$. |
If $f(0) = 0$ and $g(0) = 0$, then $f(g(p)) = (f \circ g)(p)$. |
The identity function is the identity map on polynomials. |
The identity function is the identity function on polynomials. |
If $f(x) = g(x)$ for all $x$ in the set of coefficients of a polynomial $p$, then $f(p) = g(p)$. |
If $f(0) = 0$, then $f(x^n) = f(x)^n$. |
If $f$ is the identity function on the coefficients of $p$, then $f(p) = p$. |
If $f$ is the identity function on the coefficients of $p$, then $p = f(p)$. |
The scalar multiple of a polynomial $p$ by a constant $c$ is the same as the polynomial obtained by multiplying each coefficient of $p$ by $c$. |
The polynomial $[:n:]$ is equal to the natural number $n$. |
The polynomial $n$ is equal to the monomial $nx^0$. |
The degree of a natural number is zero. |
The leading coefficient of a polynomial with integer coefficients is the integer itself. |
The polynomial $[:k:]$ is equal to the integer $k$. |
$\mathbb{Z} \subseteq \mathbb{Q}[x]$. |
The degree of an integer is zero. |
The leading coefficient of an integer is itself. |
The polynomial $[:n:]$ is equal to the numeral $n$. |
The numeral $n$ is equal to the monomial $x^0$ with coefficient $n$. |
The degree of a numeral is zero. |
The leading coefficient of a polynomial with a single term is the coefficient of that term. |
If $p$ divides $q$, then $p$ divides $aq$. |
If $p$ divides $aq$ and $a \neq 0$, then $p$ divides $q$. |
If $a$ is a nonzero element of a field, then $p$ divides $aq$ if and only if $p$ divides $q$. |
If $a \cdot p$ divides $q$, then $p$ divides $q$. |
If $p$ divides $q$, then $ap$ divides $q$. |
A polynomial $p$ divides a polynomial $q$ if and only if $p$ divides $q$ or $q = 0$. |
A polynomial $p$ is a unit if and only if $c p$ is a unit for all $c \in \mathbb{Z}$. |
If $p$ and $q$ are nonzero polynomials, then the degree of their product is the sum of their degrees. |
If $f_1, \ldots, f_n$ are nonzero polynomials, then the degree of their product is the sum of their degrees. |
The degree of the product of two polynomials is zero if and only if one of the polynomials is zero or both polynomials are constants. |
If $p$ is a nonzero polynomial, then the degree of $p^n$ is $n$ times the degree of $p$. |
If $q$ is a nonzero polynomial, then the degree of $p$ is less than or equal to the degree of $pq$. |
The coefficient of the leading term of the product of two polynomials is the product of the coefficients of the leading terms of the two polynomials. |
If $p$ divides $q$ and $q \neq 0$, then the degree of $p$ is less than or equal to the degree of $q$. |
If $p$ divides $q$, then either $p$ has degree less than or equal to $q$ or $q = 0$. |
A constant polynomial $c$ divides a polynomial $p$ if and only if $c$ divides every coefficient of $p$. |
A constant polynomial divides another constant polynomial if and only if the constant divides the other constant. |
The leading coefficient of the product of two polynomials is the product of the leading coefficients of the two polynomials. |
The leading coefficient of the product of a set of polynomials is the product of the leading coefficients of the polynomials. |
The leading coefficient of a polynomial is multiplicative under scalar multiplication. |
The leading coefficient of the polynomial $1$ is $1$. |
The leading coefficient of a polynomial raised to a power is the leading coefficient of the polynomial raised to that power. |
A polynomial is positive if and only if either it is positive except for the leading coefficient, or it is the zero polynomial and the leading coefficient is positive. |
The zero polynomial is not positive. |
If $p$ and $q$ are positive polynomials, then $p + q$ is a positive polynomial. |
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