Skip to content

Problem 1: Polynomial (100pts)

Problem

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 \(0\).

在数学中，多项式是一个由不定元（也称为变量）和系数组成的表达式，它只涉及变量的加法、减法、乘法和非负整数幂运算。一个单不定元 \(x\) 的多项式示例如 \(x^2 − 4x + 7\)。

我们可以使用 Polynomial 类来表示多项式，并使用一个数字列表来表示系数。例如，Polynomial([a_0, a_1, ..., a_n]) 表示多项式 \(a_0 + a_1x + \dots + a_nx^n\)。

请实现 Polynomial 类的所需方法，使得表示一个 Polynomial 对象 Polynomial([a_0, a_1, ..., a_n]) 时显示 Polynomial([a_0, a_1, ..., a_n])，打印它时显示 \(a_0 + a_1 \cdot x^1 + \dots + a_n \cdot x^n\)，并且我们可以直接对两个多项式对象进行加法或乘法运算。

注意：多项式的最高次系数不应为 \(0\)，除非该多项式本身就是 \(0\)。

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 ***"

Hints

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.

  • 你可以使用 str 函数将数字转换为字符串，并可以使用 + 将字符串组合起来。

  • 你可能会发现 Python 内建函数 enumerate 会有所帮助。