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
Which expression is NOT a factor of \( 1,120x^4-43,750 \)?
原库参考解析
多项式・因式与余式挑战
One of the factors of \( 8x^4+26x^2+21 \) is \( ax^2+b \), where a and b are positive constants. Which of the following is a possible value of ab?
参考答案
多项式・因式与余式进阶
The table shows several values of x and their corresponding values of \( f(x) \) for the polynomial function \( f \). Which of the following could define \( f \)?
参考答案
多项式・因式与余式进阶
\( f(x)=11x^{10}+2x^8 \)
\( g(x)=−14x^7+9x^5 \)
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 \( p(x) \)?
参考答案
多项式・因式与余式进阶
When the polynomial function g is graphed in the xy-plane such that \( y=g(x) \), it passes through the points (6, 0), (7, 0), and (0, 13). Which of the following must be true?
I. When \( g(x) \) is divided by (x − 6), 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 − 13), the remainder is 0.
参考答案
多项式・因式与余式挑战
Which expression is a factor of \( y^2(x−8)−16(x−8)^3 \)?
参考答案
多项式・因式与余式挑战
\( f(x)=13x^{10}+4x^8 \)
\( g(x)=-19x^7+8x^5 \)
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 \( p(x) \)?
\( g(x)=-19x^7+8x^5 \)
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 \( p(x) \)?
参考答案
多项式・因式与余式挑战
The function h is defined by \( h(x)=(x+p)(x-4)(2x-12) \), where p is a constant. In the xy-plane, the graph of \( y=h(x) \) passes through the point (-2, 0). What is the value of h(0)?
参考答案
多项式・因式与余式进阶
\( 4x(x-3)(x+8)=0 \)
What is one of the solutions to the given equation?
What is one of the solutions to the given equation?
原库参考解析
多项式・因式与余式挑战
\( (5x+10)(2x+1)−(3x+34) \)
Which of the following is a factor of the given expression ?
I. \( 2x+6 \)
II. \( 10x−8 \)
Which of the following is a factor of the given expression ?
I. \( 2x+6 \)
II. \( 10x−8 \)
参考答案
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 b?
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 b?
参考答案
多项式・因式与余式进阶
\( 2x(x-1)(x+7)=0 \)
What is one of the solutions to the given equation?
What is one of the solutions to the given equation?
参考答案
