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 4. What is the value of a ?
原库参考解析
多项式・因式与余式进阶
The graph of \( y=f\left(x\right) \) is shown, where the function \( f \) is defined by \( f\left(x\right)=ax^3+bx^2+cx+d \) and a, b, c, and d are constants. For how many values of x does \( f\left(x\right)=0 \)?
原库参考解析
多项式・因式与余式进阶
\( 52\left(x^3+64\right)\left(x^4-81\right)=0 \)
How many distinct real solutions does the given equation have?
原库参考解析
多项式・因式与余式基础
The graph of the polynomial function \( f \), where \( y=f\left(x\right) \), is shown. The y-intercept of the graph is \( \left(0,\ y\right) \). What is the value of y?
原库参考解析
多项式・因式与余式基础
For the polynomial function \( f \), the graph of\( y=f\left(x\right) \)in the xy-plane passes through the points (-5,0), (1,0), and (4,0). Which of the following must be a factor of \( f\left(x\right) \) ?
原库参考解析
多项式・因式与余式挑战
\( (9x^4+8x^3)-(7x^3+5x^2)-(3x^2+x) \)
The given expression can be written in the form \( 9x^4+x^3+ax^2-x \), where a is a constant. What is the value of a?
原库参考解析
多项式・因式与余式进阶
The graph of y = f(x) is shown, where the function f is defined by f(x) =ax3+ bx2+ cx + d and a,b,c, and d are constants.For how many values of x does f(x) = 0?
原库参考解析
多项式・因式与余式进阶
Which of the following could be the equation of the graph shown in the xy-plane?
原库参考解析
多项式・因式与余式挑战
One of the factors of 2x3 ＋42x2+208x is x + b, where b is a positive constant. What is the smallest possible value of b?
原库参考解析
多项式・因式与余式进阶
(x+ 2)(x-5)(x+9)=0
What is a positive solution to the given equation?
原库参考解析
多项式・因式与余式挑战
The function g is defined by \( g(x)=x(x−2)(x+6)^{2} \) . The value of g(7 - w) is 0, where w is a constant. What is the sum of all possible values of w?
原库参考解析
多项式・因式与余式进阶
What is an x-coordinate of an x-intercept of the graph of \( y=3\left(x-14\right)\left(x+5\right)\left(x+4\right) \) in the xy-plane?
原库参考解析
