用字典实现Polynomial

可以使用 Python 中的字典来实现多项式。我们可以把每个多项式的指数作为字典的键，系数作为键对应的值。例如，多项式 `3x^2 + 5x + 2` 可以表示为 `{2: 3, 1: 5, 0: 2}`。

下面是一个实现 Polynomial 类的示例代码：

class Polynomial:
    def __init__(self, coeffs):
        self.coeffs = coeffs

    def __str__(self):
        terms = []
        for exp, coeff in sorted(self.coeffs.items(), reverse=True):
            if coeff == 0:
                continue
            elif exp == 0:
                terms.append(str(coeff))
            elif coeff == 1:
                terms.append(f'x^{exp}')
            else:
                terms.append(f'{coeff}x^{exp}')
        return ' + '.join(terms)

    def __add__(self, other):
        res = {}
        for exp, coeff in self.coeffs.items():
            res[exp] = coeff
        for exp, coeff in other.coeffs.items():
            res[exp] = res.get(exp, 0) + coeff
        return Polynomial(res)

    def __sub__(self, other):
        res = {}
        for exp, coeff in self.coeffs.items():
            res[exp] = coeff
        for exp, coeff in other.coeffs.items():
            res[exp] = res.get(exp, 0) - coeff
        return Polynomial(res)

    def __mul__(self, other):
        res = {}
        for exp1, coeff1 in self.coeffs.items():
            for exp2, coeff2 in other.coeffs.items():
                exp = exp1 + exp2
                coeff = coeff1 * coeff2
                res[exp] = res.get(exp, 0) + coeff
        return Polynomial(res)

在这个类中，`__init__` 方法接收一个字典作为参数，表示多项式的系数。`__str__` 方法将多项式转换为字符串表示形式，例如 `3x^2 + 5x + 2`。`__add__` 和 `__sub__` 方法分别实现了多项式的加法和减法。`__mul__` 方法实现了多项式的乘法。这些方法都返回一个新的 Polynomial 对象，以便支持链式调用。

这里是一个使用示例：

p1 = Polynomial({2: 3, 1: 5, 0: 2})
p2 = Polynomial({1: 2, 0: 1})
print(p1)  # 输出 "3x^2 + 5x + 2"
print(p2)  # 输出 "2x + 1"
print(p1 + p2)  # 输出 "3x^2 + 7x + 3"
print(p1 - p2)  # 输出 "3x^2 + 3x + 1"
print(p1 * p2)  # 输出 "6x^3 + 11x^2 + 12x + 2"