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
For a polynomial function f , the graph of \( y=f\left(x\right) \) in the xy-plane contains the points(\( −6,0 \)), (\( 7,\ 0 \) ), (\( 0,\ 0 \) ), 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
For the polynomial function f, a solution to the equation \( f\left(x\right)=0 \) is \( x=7 \) , and one of the factors of \( f\left(x\right) \) is \( x+5 \) . Which of the following could be the graph of \( y=f\left(x\right) \) in the xy-plane?
原库参考解析
多项式・因式与余式挑战
For the polynomial function v, \( v\left(2\right)=3 \) and \( v\left(0\right)=-13 \). The graph of \( y=v\left(x\right) \) in the xy-plane passes through the points \( \left(-2,7\right) \) and \( \left(6,0\right) \). Which of the following must be true?
原库参考解析
多项式・因式与余式挑战
\( 2x(x−4)(x+9)=0 \)
What is one of the solutions to the given equation?
原库参考解析
多项式・因式与余式进阶
The expression \( \left(6x+y\right)\left(x+6y\right)\left(4x+y\right) \)is equivalent to \( ax^{3}+bx^{2}y+c \)\( xy^{2}+dy^{3} \) ，where a, b, c, and d are constanuts. What is the value of c?
原库参考解析
多项式・因式与余式挑战
\( 34z^{18}+bz^9+30 \)
In the given expression, b is a positive integer. If \( qz^9+r \) 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・因式与余式挑战
Exactly how many x-intercepts does the graph of \( f\left(x\right)=x\left(x-10\right)^2\left(x-11\right)^3 \) have in the xy-plane, where \( y=f\left(x\right) \)?
原库参考解析
多项式・因式与余式进阶
The graph of the polynomial function \( f \) in the xy-plane, where \( y=f\left(x\right) \), has x-intercepts of \( \left(-2,\ 0\right) \) and \( \left(8,\ 0\right) \). Which of the following must be true?
原库参考解析
多项式・因式与余式进阶
\( \left(x-20\right)\left(x-18\right)\left(x+9\right)\left(x+17\right)=0 \)
What is a positive solution to the given equation?
原库参考解析
多项式・因式与余式进阶
\( -5\left(v-8\right)\left(v-2\right)\left(v+9\right)=0 \)
What value of v is a solution to the given equation?
原库参考解析
多项式・因式与余式挑战
The function h is defined by \( h\left(x\right)=\left(x+p\right)\left(x-4\right)\left(2x-6\right) \), where p is a constant. In the xy-plane, the graph of \( y=h\left(x\right) \) passes through the point \( \left(-6,\ 0\right) \). What is the value of \( h\left(0\right) \)?
原库参考解析
多项式・因式与余式进阶
\( \left(x-16\right)\left(x-12\right)\left(x+7\right)\left(x+13\right)=0 \)
What is a positive solution to the given equation?
原库参考解析
