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
One of the factors of \( 5x^3+105x^2+540x \)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
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 x3 + 2x2 − 23x − 60 is x + k, where k is a constant. What is the smallest possible value of k ?
原库参考解析
多项式・因式与余式挑战
Which of the following expressions has a factor of x ＋ 2b, where b is a positive integer constant?
原库参考解析
多项式・因式与余式进阶
f(x) = x + 4
g(x) =3x2 - rx + 48
The functions f and g are given. In function g, r is a constant. If f(x) · g(x) = 3x3+192, what is the value of r ?
原库参考解析
多项式・因式与余式进阶
34z14 + bz7 + 70
In the given expression, b is a positive integer. If qz7 + r is a factor of the expression, where q and r are positive integers, what is the greatest possible value of b?
原库参考解析
多项式・因式与余式进阶
The graph of the function f in the xy-plane, where y = f(x), has x-intercepts of (-5,0),(3,0), and (6,0). Which of the following must be true?
原库参考解析
多项式・因式与余式挑战
Which of the following expressions has a factor of x ＋2b, where b is a positive integer constant?
原库参考解析
多项式・因式与余式进阶
The function f is defined by f (x)=(x - 5)(x - 9)( x + k), where k is a constant. In the xy-plane, the graph of y = f (x) passes through the point (-3, 0). What is the value of f (0)?
原库参考解析
多项式・因式与余式进阶
The function f is defined by \( f\left(x\right)=\left(x-4\right)\left(x-11\right)\left(x+k\right) \) , where k is a constant. In the xy-plane, the graph of \( y=f\left(x\right) \) passes through the point\( \left(-2,0\right) \) . What is the value of f \( \left(0\right) \) ?
原库参考解析
多项式・因式与余式挑战
Which of the following expressions is equivalent to \( 11x^{10}-11x^9+77x \) ？
原库参考解析
多项式・因式与余式进阶
\( 2x\left(x-3\right)\left(x+5\right)=0 \)
What is one of the solutions to the given equation?
原库参考解析
