2. 在本题中必须展示所有解题步骤。依赖计算器技术的解答不可接受。

已知 $x = 2 + 3\mathrm{i}$ 是方程

的一个根，

(a) 写出该方程的另一个复根。

(b) 用代数方法确定方程的另外两个根。

(c) 在同一幅 Argand 图上画出所有 4 个根。

多项式的系数全为实数，因此非实复根必成共轭对出现。把 $2+3\mathrm{i}$ 的虚部变号即可。

Since the polynomial has real coefficients, complex roots occur in conjugate pairs. Therefore another root is

先把共轭复根对应的两个一次因式相乘，得到实系数二次因式。再用多项式除法从原四次式中除去该因式，最后解余下的二次方程，并保持精确形式。

The two known roots give a quadratic factor $F(x)$, where

Let $P(x)$ denote the polynomial in the question. To divide $P(x)$ by $F(x)$, the first term of the quotient is $2x^2$:

Subtracting gives

The next term of the quotient is therefore $3$, and the remainder is zero. Thus

That is,

Hence the other roots satisfy

Thus

so the other two roots are

也可以先用两根的和与积构造二次因式，再设另一个二次因式的待定系数。展开并比较系数，可以不写多项式长除法而得到相同的剩余因式。

The sum and product of the two known roots are

and

Therefore their quadratic factor is

Let $P(x)$ denote the polynomial in the question, and write

Expanding the product,

Comparing the coefficients of $x^3$ and the constant terms gives

and

Hence $p=0$ and $q=3$. The remaining coefficients also agree, since

and

Thus the remaining factor is $2x^2+3$. Therefore,

在 Argand 图中，横坐标是实部，纵坐标是虚部。两根 $2\pm3\mathrm{i}$ 位于直线 $\operatorname{Re}(z)=2$ 上；另两根为纯虚数，位于虚轴上，并且关于实轴成对称分布。

The four roots are represented by the points