将多项式 $2x^4+ax^3+4x^2+bx-3$ 记为 $\mathrm{p}(x)$。

已知 $(x^2+x+1)$ 是 $\mathrm{p}(x)$ 的一个因式。

(a) 求 $a$ 和 $b$ 的值。

(b) 由此证明 $(x+3)$ 是 $\mathrm{p}(x)$ 的一个因式。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

逐次除以二次因式。余式必须为零，便得到两个未知系数的方程。

Dividing $p(x)$ by $x^2+x+1$, the first quotient term is $2x^2$. After subtraction, the remainder is

The next quotient term is $(a-2)x$. Subtracting its product with the divisor leaves

The final quotient term is $4-a$, leaving the remainder

Since $x^2+x+1$ is a factor, this remainder is identically zero. Hence

and the quotient is $2x^2+5x-3$.

设二次商并比较系数，最高次项和常数项可先确定商的首尾两项。

Write

Expanding,

Comparing coefficients,

Therefore $a=c+2=7$ and $b=c-3=2$, so

用因式定理代入 $x=-3$，明确算出零。

Using $a=7$ and $b=2$,

Hence, by the factor theorem, $\boxed{x+3\text{ is a factor of }p(x)}$.

利用(a)得到的商，把它继续分解。

From (a),

Since

we obtain

Thus $\boxed{x+3\text{ is a factor of }p(x)}$.