Multiplication distributes over subtraction and addition?

Answer 1

The answer to the question: Is [elementary] algebraic addition distributive over multiplication ? is generally NO of course. The word "generally" implies that the answer could be YES in certain circumstances. [The distribution of multiplication over addition may be taken as always true.]

Where the subject of this question is true, then it is found that a relationship exists between the algebraic variables. Consider the equation:

a + b.c = [a + b].[a + c] [This equation is not generally true as is found by substituting any values for a, b and c.] However multiplying out the right side gives: a + b.c = a**{2} + a.c + b.a + b.c Cancelling the term b.c enables us to solve for the variable: a in terms of the variables: b and c. We find that:

a = 1 - b - c . There are clearly an infinite number of possible values of a for pairs of b and c.

Note that if b and c are uneven integers, then a is also an uneven integer.

Check: set b = 5, c = 7, then a = - 11 from the answer given.

Now a + b.c = - 11 + 35 = 24 and [a + b] = - 6 and [a + c] = - 4 so that:

[a + b].[a + c] = [- 6].[- 4] = 24 .

A similar result can be found for the distribution of subtraction over multiplication and moreover these examples can be made much more general.

In the case of subtraction, an uneven number of factors are needed to cancel the multiplicative term b.c.d on the left, thus:

a - b.c.d = [a - b].[a - c].[a - d]