f(x) = x 3 + 2x 2 + x 的倍数零和重数是多少？
数制是一种在数轴上表示数字的方法。符号范围为 0-9，称为数字。多项式是函数 (x) = a n x n + a n−1 x n−1 + ... + a 2 x 2 + a 1 x + a 0形式的函数。多项式的次数是表达式中 x 的最高幂。常数（非零）多项式为 0 次，线性多项式（x 的最大幂为 1）为 1 次，二次多项式（x 的最大幂为 2）为 2 次，依此类推。
根或零
如果 a 和 b 是根，那么具有这些根的多项式函数是 f(x) = (x – a)(x – b)，或者是这个的倍数。例如，如果二次表达式具有根 x = 3 和 x = -2，则函数必须是 f(x) = (x – 3)(x + 2)，或者是它的常数倍。这可以应用于任何次数的多项式。例如，如果多项式的根是 x = 2、x = 3、x = 4，则函数必须是 f(x) = (x – 2)(x – 3)(x – 4)，或者这个的恒定倍数。让我们也尝试考虑函数f(x) = (x – 1) 2 。可以看出x – 1 = 0，所以x = 1。对于这个函数，有一个根。这就是所谓的重复根，这个根可以重复任意次数。例如，f(x) = (x – 2) 3 (x + 4) 4有一个重复根 x = 2，一个重复根 x = -4。可以说，根 x = 2 的重数为 3，而根 x = -4 的重数为 4。
多重性和多重根
多项式的多个根是其因子在多项式的完全因式分解中多次出现的根。我们将一个因子在完全因式分解中出现的次数称为根的多重性。以下示例将演示如何找到多重性和多重根。
f(x) = x 3 + 2x 2 + x 的倍数零和重数是多少？
解决方案：
The Objective here to is to find the Multiple zero and multiplicity of f(x) = x3 + 2x2 + x. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. A multiple zero is a root with multiplicity m ≥ 2.
f(x) = x3 + 2x2 + x. Will be equated to zero.
x3 + 2x2 + x = 0
x(x2 + 2x + 1) = 0 (extract x common from the equation and the remaining part becomes a quadratic equation)
x2 + 2x + 1 can be written as (x + 1)2 it can be seen that the roots or zeroes of f(x) are 0, -1. Here zero has a multiplicity of 1 since it occurs once in the factored form. -1 has a multiplicity of 2. Therefore, multiple zero of f(x) = x3 + 2x2 + x, is -1 and it has multiplicity of 2.
类似问题
问题 1：y = 3(x + 3) 3 (x + 2) 4 (x – 1) 2 (x – 5) 的倍数零和重数是多少。
解决方案：
Roots of this function are,
x + 3 = 0 -> x = -3
x + 2 = 0 -> x = -2
x – 1 = 0 -> x = 1
x – 5 = 0 -> x = 5
Multiple zeroes are -5, -2, 1. Multiplicity of x = -5 is 3 because x + 5 is raised to the power 3, Similarly, x = -2 is 4 and x = 1 is 2.
问题2：y = (x + 1) 2 (x + 3) 3的倍数零和重数是多少
解决方案：
Roots of this function are,
x + 1 = 0 -> x = -1
x + 3 = 0 -> x = -3F
Multiple zeroes are -1, 3. The multiplicity of x = -1 is 2 because x + 1 term is raised to the power 2 and multiplicity of x = -3 is 3 because x + 3 is raised to the power 3