Why can’t you distribute when multiplying monomials?

Question

My understanding is that you can simplify the expression 3x(4x) by multiplying the coefficients 3 * 4 = 12 and the variables x * x = x² so the answer is 12x². But if I instead represent 3x(4x) as (3*x)(4*x), intuition tells me that I could distribute and simplify like this

(3*x)(4*x)

4(3*x) + x(3*x)

4*3 + 4x + 3x + x²

12 + 7x + x²

But 12 + 7x + x² and 12x² are not equivalent. Why does distributing not work?

Answer 1

I believe the answer lies in the understanding of what a term is. A term would need to be separated by a an +/- sign or you have a monomial ( single term). In this case both 3x and 4x or monomials. Distribution would only occur if one of the multipliers had more than one term.

Answer 2

This is actually a fairly common question among algebra students. The simplest answer is that the distributive property only applies to summations, not products. For example,

c*(a+b) = c*a + c*b, you can verify this by plugging in actual numbers:

2*(3+1) = 2*3+2*1 = 8 <- This is correct

If you try to do this with products, it doesn't work:

2*(3*1) = 2*3 + 2*1 = 8 <- This is NOT correct

The same logic applies to monomial multiplication. Just remember that if you end up with a plus sign that wasn't there when you started, something is probably wrong.