An equation of degree n has n roots. Multiple roots are the repetitions of these n roots. For example, if the square of x is equal to 0, the multiplicity is 1 and both roots are equal to 0. There is a formula: degree/multiplicity = how many roots with different values. So the multiplicity must be the divisor of the number.

According to the basic theorem of algebra, P(x) in complex number field can always be decomposed into the product of linear terms. In the decomposition formula of P(x), the degree of (x-t) is the multiplicity of the root x = t. For example, (x- 1) 3 * (x-5) = 0, 1 is a triple root, and 5 is 1.

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For algebraic equations, that is, polynomial equations, if the equation f(x) = 0 has roots and x = a, it means that f(x) has a factor (x-a), so we do polynomial division and the result is still polynomial. If P(x) = 0 still takes x= a as the root, then x = a is the multiple root of the equation. Or let f 1(x) be the derivative of f(x). If f 1(x) = 0 is also based on x =a, it can also be explained that x= a is the multiple root of the equation f(x)=0.

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