第一题： 对如下多项式（Polynomial）编写类定义：其中，n为多项式的次数。完成如下功能： （1）可存储任意大的多项式（提示：可用动态数组实现）。 （2）定义构造函数、析构函数、拷贝构造函数。 （3）包含一个static成员存储定义的多项式数量。 （4）定义一个成员函数输出多项式。（可参照-x^4-6x^3+5格式输出，注意化简） （5）定义一个成员函数计算多项式的值。 （6）重载“+”运算符，实现两个多项式相加。 （7）重载“-”运算符，实现两个多项式相减。 （8）重载“*”运算符，实现两个多项式相乘。 （9）重载“=”运算符，实现两个多项式的赋值运算。 （10）在main函数中编写测试代码。 要求：采用多文件实现。考虑：哪些成员函数可以声明为const， 把其中某个运算符重载为友元函数。

在头文件 Polynomial.h 中编写类定义：

#ifndef POLYNOMIAL_H
#define POLYNOMIAL_H

class Polynomial {
private:
    int n; // 多项式的次数
    double *coefficients; // 多项式系数数组
    static int count; // 定义的多项式数量
public:
    Polynomial(); // 默认构造函数，创建一个0次多项式
    Polynomial(int n, double *coefficients); // 指定次数和系数数组的构造函数
    Polynomial(const Polynomial &p); // 拷贝构造函数
    ~Polynomial(); // 析构函数
    void output(); // 输出多项式
    double calculate(double x); // 计算多项式的值
    Polynomial operator+(const Polynomial &p) const; // 重载+运算符
    Polynomial operator-(const Polynomial &p) const; // 重载-运算符
    Polynomial operator*(const Polynomial &p) const; // 重载*运算符
    Polynomial &operator=(const Polynomial &p); // 重载=运算符
    static int getCount(); // 获取定义的多项式数量
};

#endif

在源文件 Polynomial.cpp 中实现成员函数：

#include "Polynomial.h"
#include <iostream>

using namespace std;

int Polynomial::count = 0; // 初始化定义的多项式数量为0

Polynomial::Polynomial() {
    n = 0;
    coefficients = new double[1];
    coefficients[0] = 0.0;
    count++;
}

Polynomial::Polynomial(int n, double *coefficients) {
    this->n = n;
    this->coefficients = new double[n + 1];
    for (int i = 0; i <= n; i++) {
        this->coefficients[i] = coefficients[i];
    }
    count++;
}

Polynomial::Polynomial(const Polynomial &p) {
    n = p.n;
    coefficients = new double[n + 1];
    for (int i = 0; i <= n; i++) {
        coefficients[i] = p.coefficients[i];
    }
    count++;
}

Polynomial::~Polynomial() {
    delete[] coefficients;
    count--;
}

void Polynomial::output() {
    if (n == 0) {
        cout << coefficients[0] << endl;
        return;
    }
    if (coefficients[0] != 0) {
        cout << coefficients[0];
        if (n != 0) {
            cout << " + ";
        }
    }
    for (int i = 1; i < n; i++) {
        if (coefficients[i] != 0) {
            cout << coefficients[i] << "x^" << n - i << " + ";
        }
    }
    if (coefficients[n] != 0) {
        cout << coefficients[n] << "x^0" << endl;
    }
}

double Polynomial::calculate(double x) {
    double result = coefficients[0];
    for (int i = 1; i <= n; i++) {
        result += coefficients[i] * pow(x, i);
    }
    return result;
}

Polynomial Polynomial::operator+(const Polynomial &p) const {
    int maxDegree = max(n, p.n);
    double *newCoefficients = new double[maxDegree + 1];
    for (int i = 0; i <= maxDegree; i++) {
        newCoefficients[i] = (i <= n ? coefficients[i] : 0) + (i <= p.n ? p.coefficients[i] : 0);
    }
    Polynomial result(maxDegree, newCoefficients);
    delete[] newCoefficients;
    return result;
}

Polynomial Polynomial::operator-(const Polynomial &p) const {
    int maxDegree = max(n, p.n);
    double *newCoefficients = new double[maxDegree + 1];
    for (int i = 0; i <= maxDegree; i++) {
        newCoefficients[i] = (i <= n ? coefficients[i] : 0) - (i <= p.n ? p.coefficients[i] : 0);
    }
    Polynomial result(maxDegree, newCoefficients);
    delete[] newCoefficients;
    return result;
}

Polynomial Polynomial::operator*(const Polynomial &p) const {
    int newDegree = n + p.n;
    double *newCoefficients = new double[newDegree + 1];
    for (int i = 0; i <= newDegree; i++) {
        newCoefficients[i] = 0;
    }
    for (int i = 0; i <= n; i++) {
        for (int j = 0; j <= p.n; j++) {
            newCoefficients[i + j] += coefficients[i] * p.coefficients[j];
        }
    }
    Polynomial result(newDegree, newCoefficients);
    delete[] newCoefficients;
    return result;
}

Polynomial &Polynomial::operator=(const Polynomial &p) {
    if (this != &p) {
        n = p.n;
        delete[] coefficients;
        coefficients = new double[n + 1];
        for (int i = 0; i <= n; i++) {
            coefficients[i] = p.coefficients[i];
        }
    }
    return *this;
}

int Polynomial::getCount() {
    return count;
}

在主函数中编写测试代码：

#include "Polynomial.h"
#include <iostream>

using namespace std;

int main() {
    double c1[] = {3, 2, 1};
    Polynomial p1(2, c1);
    cout << "p1: ";
    p1.output(); // 输出 p1

    double c2[] = {4, 5, 6};
    Polynomial p2(2, c2);
    cout << "p2: ";
    p2.output(); // 输出 p2

    Polynomial p3 = p1 + p2;
    cout << "p3 = p1 + p2: ";
    p3.output(); // 输出 p3

    Polynomial p4 = p1 - p2;
    cout << "p4 = p1 - p2: ";
    p4.output(); // 输出 p4

    Polynomial p5 = p1 * p2;
    cout << "p5 = p1 * p2: ";
    p5.output(); // 输出 p5

    Polynomial p6;
    p6 = p1;
    cout << "p6 = p1: ";
    p6.output(); // 输出 p6

    cout << "p1(1) = " << p1.calculate(1) << endl; // 计算 p1 在 x=1 处的值

    cout << "已定义多项式数量：" << Polynomial::getCount() << endl; // 输出已定义多项式数量

    return 0;
}