这部分需要会什么？
- 用零乘积性质求多项式零点
- 理解因式定理和余式定理
- 根据因式形式识别横截距与重根
核心概念与公式
若多项式 \(P(x)\) 含因式 \((x-r)\)，则 \(P(r)=0\)，\(r\) 是零点；反过来也成立。\(P(x)\) 除以 \((x-r)\) 的余数为 \(P(r)\)。除数是 \((ax-b)\) 且 \(a\ne0\) 时，应代入 \(x=b/a\)。
若 \(y=a(x-r_1)(x-r_2)\cdots\)，非零常数 \(a\) 不影响零点。图象的横截距为 \((r_i,0)\)。零点重数为奇数时图象穿过横轴，重数为偶数时图象在该处接触横轴而不穿过。
做题时怎样使用
- 求横截距时令 \(y=0\)，观察现成的因式，通常不必展开
- 逐个将一次因式设为零；求余数则直接代入使除数为零的输入
- 读清是要求一个零点、全部零点，还是截距坐标
容易混淆的地方
\((x+5)\) 对应零点 \(-5\)。前面的系数 4 不是横截距；“an x-coordinate”只要求一个符合条件的横坐标。
用一个例题串起方法
知识点教学例题\(y=4(x-13)(x+5)(x+1)\)。图象的横截距的横坐标有哪些？
展开解题思路 ＋
令每个一次因式为零：\(x-13=0\)、\(x+5=0\)、\(x+1=0\)。得到 \(x=13,-5,-1\)。若只要求一个横坐标，填写其中任意一个；若要求所有截距，则为 \((13,0),(-5,0),(-1,0)\)。
本知识点题库
先独立作答，再查看答案与解析。
多项式・因式与余式挑战
\( g(x)=x^3+ax^2+bx+c \)
The function g is defined by the given equation, where a, b, and c are integer constants. The zeros of the function are\( −7 \), \( −2 \), and \( 3 \). What is the value of a?
The function g is defined by the given equation, where a, b, and c are integer constants. The zeros of the function are\( −7 \), \( −2 \), and \( 3 \). What is the value of a?
参考答案
多项式・因式与余式挑战
The function h is defined by \( h(x)=(x+p)(x-2)(2x-12) \), where p is a constant. In the xy-plane, the graph of \( y=h(x) \) passes through the point \( (-7,\ 0) \). What is the value of \( h(1) \)?
参考答案
多项式・因式与余式进阶
For the polynomial function r, the graph of \( y=r(x) \) in the xy-plane passes through the points (6, 8) and (14, 0). Which of the following must be a factor of \( r(x) \)?
参考答案
多项式・因式与余式挑战
\( y=-\frac{1}{9}(x+13)(x^2-25) \)
In the xy-plane, the graph of the given equation contains the point (s, t). If \( -5\le s\le5 \), which of the following is NOT a possible value of t?
In the xy-plane, the graph of the given equation contains the point (s, t). If \( -5\le s\le5 \), which of the following is NOT a possible value of t?
参考答案
多项式・因式与余式挑战
Which expression is a factor of \( y^2(x-6)-16(x-6)^3 \)?
参考答案
多项式・因式与余式挑战
\( y=-\frac{1}{8}(x+13)(x^2-49) \)
In the xy-plane, the graph of the given equation contains the point (s, t). If \( -7\le s\le7 \), which of the following is NOT a possible value of t?
In the xy-plane, the graph of the given equation contains the point (s, t). If \( -7\le s\le7 \), which of the following is NOT a possible value of t?
参考答案
多项式・因式与余式挑战
Which expression is a factor of \( y^2(x-8)-4(x-8)^3 \)?
参考答案
多项式・因式与余式进阶
The binomial \( x+6y \) is a factor of which of the following?
参考答案
多项式・因式与余式进阶
Which expression is a factor of \( 56y^3+40py^2+21y+15p \)?
参考答案
多项式・因式与余式进阶
\( y=x^3+bx^2+260x-372 \)
In the xy-plane, the graph of the given equation, where b is a constant, contains the points (2, 0), (r, 0), and (6, 0). What is the value of r?
In the xy-plane, the graph of the given equation, where b is a constant, contains the points (2, 0), (r, 0), and (6, 0). What is the value of r?
参考答案
多项式・因式与余式挑战
The graph of \( y=f(x) \) is shown where \( f(x)=ax^3+bx^2+cx+d \) and a, b, c, and d are constants. For how many distinct values of x does \( f(x)=0 \)?
参考答案
多项式・因式与余式挑战
Which expression is a factor of \( 36x^2\left(x+12\right)-36\left(x+12\right)^3 \) ?
原库参考解析
