Polynomials¶

class ruffini.Polynomial(terms=(), *args)¶

Bases: tuple

A Polynomial object is the sum of two or more monomials and/or numbers.

You can sum, subtract and multiplicate instances of Polynomial.

You can assign a value to the variables and calculate the value of that polynomial with the value you assigned.

NB The Polynomial class is a subclass of tuple, so all the methods of tuple are automatically inherited from Polynomial; many of these methods are not in this docs.

static __new__(cls, terms=(), *args)¶

Create the polynomial by giving it a list of terms (a term can be a Monomial or a number); if two or more terms have the same variables, it will sum them toghether

>>> Polynomial(Monomial(2, x=2, y=2), Monomial(3, x=2, y=2))
5x**2y**2

Raise:	TypeError

__init__(terms=(), *args)¶

Initialize the polynomial, then calculate its degree (the highest degree between terms’ ones)

>>> p = Polynomial(Monomial(a=1), Monomial(3))
>>> p
a + 3
>>> p.degree
1

term_coefficient(variables=None, **kwargs)¶

Return the coefficient of the term with the given variables

>>> x = Variable('x')
>>> y = Variable('y')
>>>
>>> p = 2*x*y - 6*x*y**3 + 8*x*y
>>> p
10xy - 6xy**3
>>>
>>> p.term_coefficient(x=1, y=1)
Fraction(10, 1)

If none is found, the result will be 0

>>> p.term_coefficient(k=1, b=2)
0

You can also give directly the variables as an argument

>>> p.term_coefficient(x*y)
Fraction(10, 1)

Return type:	int, float

factorize()¶

With this method you can factorize the polynomial.

For more informations, see factorize() docs.

Return type:	FPolynomial

zeros¶

Return a set of zeros for the polynomial

>>> x = Variable('x')
>>> p = 3*x**3 + 2*x**2 - 3*x - 2
>>> p.zeros
{Fraction(1, 1), Fraction(-2, 3), Fraction(-1, 1)}

It works only with polynomials with only a variable and a constant term

>>> Polynomial(3*x, Monomial(2, y=1)).zeros
Traceback (most recent call last):
...
ValueError: Can't calculate zeros for polynomials with more than a variable

>>> Polynomial(3*x, 5*x**2).zeros
Traceback (most recent call last):
...
ValueError: Can't calculate zeros for polynomials without a constant term

Return type:	set
Raises:	ValueError

eval(values={}, **kwargs)¶

Evaluates the polynomial, giving values for each variable

>>> p = Polynomial(Monomial(5, x=1), Monomial(3, y=1))
>>> p.eval(x=2, y=3)
19

For more informations, see Monomial.eval().

Return type:	int, float, Monomial, Polynomial

__add__(other)¶

Sum the polynomial with another polynomial, a monomial or a number, too.

>>> x = Variable('x')
>>> y = Variable('y')
>>> p = 3*x + 2*y
>>>
>>> p + (3*y + 2)
3x + 5y + 2
>>>
>>> p + 2*x
5x + 2y
>>>
>>> p + 1
3x + 2y + 1

Return type:	Polynomial
Raise:	TypeError

__sub__(other)¶

Subtract the polynomial from another polynomial, a monomial or a number.

>>> x = Variable('x')
>>> y = Variable('y')
>>>
>>> p = 3*x + 2*y
>>>
>>> p - (3*y + 2)
3x - y - 2
>>>
>>> p - 2*x
x + 2y
>>>
>>> p - 1
3x + 2y - 1

Return type:	Polynomial
Raise:	TypeError

__mul__(other)¶

This method is used to multiply a polynomial by a polynomial, a monomial or a number:

>>> x = Variable('x')
>>> y = Variable('y')
>>>
>>> p = 3*x + 2*y
>>>
>>> p * (3*y + 2)
9xy + 6x + 6y**2 + 4y
>>>
>>> p * x
3x**2 + 2xy
>>>
>>> p * 4
12x + 8y

Return type:	Polynomial
Raise:	TypeError

__radd__(other)¶

This method is the reverse for Polynomial.__add__(). With this method, you can swap the two operands of the addition:

>>> 8 + Polynomial(Monomial(4, a=2))
4a**2 + 8

For more informations, see Polynomial.__add__() docs.

Return type:	Polynomial
Raise:	TypeError

__rsub__(other)¶

This method is the reverse for Polynomial.__sub__(). With this method, you can swap the two operands of the addition:

>>> 5 - Polynomial(Monomial(7, k=1))
-7k + 5

For more informations, see Polynomial.__sub__ docs().

Return type:	Polynomial
Raise:	TypeError

__rmul__(other)¶

This method is the reverse for Polynomial.__mul__(). With this method, you can swap the two operands of the addition:

>>> 10 * Polynomial(Monomial(3.5, b=3))
35b**3

For more informations, see Polynomial.__mul__() docs.

Return type:	Polynomial, NotImplemented
Raise:	TypeError

__str__()¶

Return the polynomial as a string. Powers are indicated with **.

>>> str(Polynomial(Monomial(4, a=4, b=1)))
'4a**4b'
>>> str(Polynomial(Monomial(a=2), Monomial(-2, c=2)))
'a**2 - 2c**2'
>>> str(Polynomial(Monomial(3, x=2), Monomial(6, y=3)))
'3x**2 + 6y**3'

To see how the single terms are printed, see the Monomial.__str__() docs.

Return type:	str

__repr__()¶

Return the polynomial as a string.

>>> repr(Polynomial(Monomial(4, a=4, b=1)))
'4a**4b'

For more informations, see Polynomial.__str__().

Return type:	str

__eq__(other)¶

Check if two polynomials are equivalent, comparing each term

>>> p0 = Polynomial(Monomial(4, a=4, b=1))
>>> p1 = Polynomial(Monomial(1, a=2), Monomial(-2, c=2))
>>> p2 = Polynomial(Monomial(-2, c=2), Monomial(1, a=2))
>>>
>>> p0 == p1
False
>>> p0 == p0
True
>>> p1 == p2
True

If a polynomial has a single term, it can also be compared to a monomial

>>> Polynomial(Monomial(3, f=2)) == Monomial(3, f=2)
True

Since a monomial with no variables can be compared to a number, if a polynomial has only a term - which is a monomial with no variables - it can be compared to a number

>>> Polynomial(Monomial(7)) == 7
True

In any other case, the result will be False.

>>> Polynomial() == {1, 2, 3}
False

Return type:	bool

__neg__()¶

Return the opposite of the polynomial, changing the sign of each term of the polynomial

>>> -Polynomial(Monomial(4, x=1), Monomial(2, y=2))
-4x - 2y**2

Return type:	Polynomial