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Having watched a video on synthetic division, which stated that: "In algebra, synthetic division is a method of performing polynomial long division." I don't think that they are similar.
A quotient is the answer of a division question and the product is the answer of a multiplication question but they are the same because they are both an answer to a math problem.
They are elements of of a set which may consist of integers, real or complex numbers, polynomial expressions, matrices.
Arithmetic operations include addition, subtraction, multiplication, division. Relational operations include different comparisons between numbers (or sometimes other data types). There are six relational operations: equal, not-equal, less-than, greater-than, less-than-or-equal, greater-than-or-equal. One difference, which may help you remember the difference is that if you combine two numbers with an arithmetic operation, for example an addition, you get another number. On the other hand, if you compare two numbers, for example with "greater than", you are asking a question which will be answered with "yes" or "no" - which in computer languages are often called "true" and "false" or something similar.
Multiplication is successive Addition Division is successive subtraction