3.3 综合除法 Synthetic Division - 练习题 - 独中课程 | 线上补习

高中一 | 高级数学

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(1)
Use synthetic division to determine the quotient Q(x) and the remainder R obtained when the first polynomial is divided by the second polynomial.

用综合除法计算当第一多项式除以第二多项式除后所得的商式Q(x) 和余数R。

(a) $2x^3 - x^2 + 2x - 3$ , $x + 1$

(b) $x^4 + 3x^3 - x + 5$ , $x - 3$ [bg_collapse view=”link” color=”#4a4949″ expand_text=”Hint” collapse_text=”Closed Hint” ]没有 $x^2$ 项，记得补0。[/bg_collapse]
(c) $2 - 3x + 5x^2 - 2x^3$ , $2x + 1$ [bg_collapse view=”link” color=”#4a4949″ expand_text=”Hint” collapse_text=”Closed Hint” ]用 $-\frac{1}{3}$ 做综合除法后商式还需除2，余数不用除2[/bg_collapse]
(d) $4x^3 - 2x^2 + x - 1$ , $3x - 2$ [bg_collapse view=”link” color=”#4a4949″ expand_text=”Hint” collapse_text=”Closed Hint” ]用 $\frac{2}{3}$ 综合除法后因式还需除3，余式不用除3[/bg_collapse]

答案Answer:

(a) $2x^2 - 3x + 5$ , $-8$

(b) $x^3 + 6x^2 + 18x + 53$ , $164$

(c) $-x^2 + 3x - 3$ , $5$

(d) $\frac{4}{3}x^2+\frac{2}{9}x+\frac{13}{27}$ , $-\frac{1}{27}$

(2)
Use synthetic division to write the following expressions in the form $Q\left(x\right)+\frac{R}{x-k}$ .
用综合除法把下列各因式写成 $Q\left(x\right)+\frac{R}{x-k}$ 的模式。[bg_collapse view=”link” color=”#4a4949″ expand_text=”Hint” collapse_text=”Close Hint” ] $Q(x)$ 为商式， $R$ 为余数， $x - k$ 为题目的分母。[/bg_collapse]

(a) $\frac{x^3+5x^2+1}{x+2}$

(b) $\frac{4x^3+x^2-2x-7}{4x+1}$ [bg_collapse view=”link” color=”#4a4949″ expand_text=”Hint” collapse_text=”Close Hint” ]Hint：用 – ¼ 做综合除法后商式还需除4，余数不用除2. [/bg_collapse]

(c) $\frac{x^4-2x^2+1}{1-2x}$ [bg_collapse view=”link” color=”#4a4949″ expand_text=”Hint” collapse_text=”Close Hint” ]没有 $x^3$ 和 $x$ 项，记得补0。用½ 做综合除法后商式还需除 $-2$ ，余数不用除 $-2$ [/bg_collapse]

答案Answer:

(a) $x^2-7x+14-\frac{27}{x-k}$

(b) $x^2-\frac{1}{2}-\frac{13}{2(4x+1)}$

(c) $-\frac{1}{2}x^3-\frac{1}{4}x^2+\frac{7}{8}x+\frac{7}{16}+\frac{9}{16(1-2x)}$

(3)
If $2x^3 - ax^2- 5x - 2$ has factor $x + 1$ , find
若 $x + 1$ 为 $2x^3 - ax^2- 5x - 2$ 的因式，求

(a) the values of a,
a 之值，
[bg_collapse view=”link” color=”#4a4949″ expand_text=”Hint” collapse_text=”Close Hint” ]用 –1做综合除法，所得余数= 0.[/bg_collapse]

(b) factorize the statement.
该式因式的分解。
[bg_collapse view=”link” color=”#4a4949″ expand_text=”Hint” collapse_text=”Close Hint” ]将（a）综合除法所得商式因式分解。[/bg_collapse]

答案Answer:

(a) $1$

(b) $(2x + 1) (x - 2) (x + 1)$