Problem 1: Polynomial (100pts)

In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate $x$ is $x^2 − 4x + 7$.

We can use class Polynomial to represent polynomials, and a list of number to represent coefficients. For example, Polynomial([a_0, a_1, ..., a_n]) represents polynomial $a_0 + a_1x + ... + a_nx^n$.

Implement required methods of class Polynomial such that representing a Polynomial object Polynomial([a_0, a_1, ..., a_n]) displays Polynomial([a_0, a_1, ..., a_n]), printing it displays a_0 + a_1*x^1 + ... + a_n*x^n and we can directly add or multiply two polynomial objects.

Note: the highest degree coefficient of a polynomial should not be 0, unless it is just $0$.

Hint1: You can convert numbers to strings using the str function, and you can combine strings together using +.

Hint2: You may find python built-in function enumerate helpful.

class Polynomial:
    """Polynomial.

    >>> a = Polynomial([0, 1, 2, 3, 4, 5, 0])
    >>> a
    Polynomial([0, 1, 2, 3, 4, 5])
    >>> print(a)
    0 + 1*x^1 + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5
    >>> b = Polynomial([-1, 0, -2, 1, -3])
    >>> print(b)
    -1 + 0*x^1 + -2*x^2 + 1*x^3 + -3*x^4
    >>> print(a + b)
    -1 + 1*x^1 + 0*x^2 + 4*x^3 + 1*x^4 + 5*x^5
    >>> print(a * b)
    0 + -1*x^1 + -2*x^2 + -5*x^3 + -7*x^4 + -12*x^5 + -11*x^6 + -15*x^7 + -7*x^8 + -15*x^9
    >>> print(a)
    0 + 1*x^1 + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5
    >>> print(b) # a and b should not be changed
    -1 + 0*x^1 + -2*x^2 + 1*x^3 + -3*x^4
    >>> zero = Polynomial([0])
    >>> zero
    Polynomial([0])
    >>> print(zero)
    0
    """
    "*** YOUR CODE HERE ***"

You may find Python string formatting syntax or f-strings useful. A quick example:

>>> ten, twenty, thirty = 10, 'twenty', [30]
>>> '{0} plus {1} is {2}'.format(ten, twenty, thirty)
'10 plus twenty is [30]'

>>> feeling = 'love'
>>> course = 61
>>> f'I {feeling} {course}A!'
'I love 61A!'