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
\( -4\left(v-9\right)\left(v-2\right)\left(v+7\right)=0 \)
What value of v is a solution to the given equation?
原库参考解析
多项式・因式与余式基础
The graph of the polynomial function \( f \), where \( y=f(x) \), is shown. The y-intercept of the graph is \( (0,\ y) \). What is the value of y?
原库参考解析
多项式・因式与余式挑战
Which expression is a factor of \( y^2(x-3)-25(x-3)^3 \) ?
原库参考解析
多项式・因式与余式挑战
\( r(x)=9(x-3) \)
\( s(x)=x^3+nx^2+3nx+27 \)
For the given functions r and s, n is a constant. If \( r(x)·s(x)=9(x^{4}−81) \) , what is the value of n?
原库参考解析
多项式・因式与余式进阶
For the polynomial function \( f(x) \) , when is divided by \( x−8 \), the remainder is 9. Which of the following must be true about the graph of \( y=f(x) \) in the xy-plane?
原库参考解析
多项式・因式与余式挑战
\( f(x)=11x^{10}+3x^8 \)
\( g(x)=-17x^7+9x^5 \)
The polynomial \( p(x) \) is defined as the product of the given polynomials, \( f(x) \) and \( g(x) \). What is the coefficient of \( x^{15} \) in \( p(x) \)?
原库参考解析
多项式・因式与余式进阶
\( f(x)=x+3 \)
\( g(x)=7x^2-rx+63 \)
The function \( f \) and g are given. In function g, r is a constant. If \( f(x)·g(x)=7x^{3}+189 \), what is the value of r?
原库参考解析
多项式・因式与余式挑战
One of the factors of \( 8x^4+50x^2+63 \) is \( ax^2+b \), where a and b are positive constants. Which of the following is a possible value of ab?
原库参考解析
多项式・因式与余式进阶
What is an x-intercept of the graph of the equation \( y=(x+3)(x-11)(x+12) \) in the xy-plane?
原库参考解析
多项式・因式与余式基础
\( p(x)=x^3+4x^2+9x+12 \)
When the given function p is graphed in the xy-plane, where \( y=p(x) \), the y-intercept of the graph is \( (0,\ y) \). What is the value of y?
原库参考解析
多项式・因式与余式挑战
For the polynomial function \( f \), the graph of \( y=f(x) \) in the xy-plane passes through the points (-5, 0), (3, 0), and (7, 0). Which of the following must be a factor of \( f(x) \) ?
原库参考解析
多项式・因式与余式挑战
When the polynomial function g is graphed in the xy-plane such that \( y=g(x) \), it passes through the points (4, 0), (7, 0), and (0, 18). Which of the following must be true?
I. When \( g(x) \) is divided by \( (x−4) \), the remainder is 0.
II. The point (7,0) is an x-intercept of the graph of \( y=g(x) \).
III. When \( g(x) \) is divided by \( (x−18) \), the remainder is 0.
原库参考解析
