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・因式与余式进阶
What is an x-coordinate of an x-intercept of the graph of y =4(x -13)(x +5)(x +1) in the xy-plane?
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-2)(x-11)(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・因式与余式挑战
f(x) = x＋4
g(x) = 8x2 - rx + 128
The functions f and g are given. In function g, r is a constant. If f(x) · g(x) =8x3+512, what is the value of r?
原库参考解析
多项式・因式与余式挑战
The graph of the polynomial function f in the xy-plane, where y = f(x), has x-intercepts of (-7,0) and (2, 0). Which of the following must be true?
原库参考解析
多项式・因式与余式进阶
ax2 +176x + c
In the given expression, a and c are positive constants. If px +q is a factor of the expression, where p and q are positive constants, what is the greatest possible value of ac ?
原库参考解析
多项式・因式与余式进阶
\( f\left(x\right)=x+3 \)
\( g\left(x\right)=5x^2-rx+45 \)
The functions \( f \) and \( g \) are given. In function \( g \), \( r \) is a constant. If \( f(x)⋅g(x) \)\( =5x^3+135 \) , what is the value of \( r \)?
原库参考解析
多项式・因式与余式挑战
How many \( x \)-intercepts does the function\( f\left(x\right)=x\left(8x+3\right)^4\left(2x+1\right)^3 \) has ?
原库参考解析
多项式・因式与余式挑战
Which of the following expressions has x + 2b as a factor, where b is a positive integer?
原库参考解析
多项式・因式与余式进阶
f(x)= 3(x - 2)(x -6)(x -9)
lf the given function f is graphed in the xy-plane, where y = f(x), what is the x-coordinate of an x-intercept of the graph?
原库参考解析
多项式・因式与余式挑战
56(x3+64)(x4 -81) = 0
How many distinct real solutions does the given equation have?
原库参考解析
多项式・因式与余式挑战
38z18+bz9+30
In the given expression, b is a positive integer. If qz9+ 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
Which of the following expressions has a factor of x + 2b, where b is a positive integer constant?
原库参考解析
