The distributive property tells how to multiple across parentheses. In particular, if you are multiplying across parentheses, you multiply each term. \[a(b + c) = ab + ac\] In the above equation, the left side show \(a\) being multiplied by \(b+c.\) On the right side, \(a\) is multiplied by \(b,\) then \(a\) is multiplied by \(c,\) then the results are added.
Let's look at an example with just numbers. When \(a = 2,\) \(b = 3,\) and \(c = 5\). The left side is \(2 \cdot (3 + 5) = 2 \cdot 8 = 16.\) Notice addition was first, then multiplication on the left. On the right side, we do multiplication first. \(2 \cdot 3 + 2 \cdot 5 = 6 + 10 = 16.\) Either way, we get \(16,\) so \(2 \cdot (3 + 5) = 2 \cdot 3 + 2 \cdot 5.\)
Multiplication doesn't depend on order, so distributing from the right works too. \[(b+c)a = ba + ca\]
You are more likely to see distribution with a variable than with numbers. For example, you may start with \(3(2x+1).\) In this case, you can't add the \(2x\) and \(1\) since they are not like terms, but you can distribute the \(3.\) The result after distribution is \(6x+3.\)
Equations work both ways, so you can also go backwards. For example, both terms in \(4x+2\) have a factor of \(2,\) so we can undo the distribution and get \(2(2x+1).\) This reverse distribution is called factoring since the result of a multiplication of two factors.
The English language has a version of distribution as well. Consider the sentence "The dog is small and the cat is small." Think of dog as \(b,\) cat as \(c,\) and small as \(a.\) This is like \(ba + ca.\) We can use the distributive property to factor, \(ba + ca = (b+c)a.\) Now the sentence becomes "The dog and cat are small."
What about the sentence "There is a green cat and dog." Now we can think of green as \(a,\) cat as \(b,\) and dog as \(c.\) A green cat is weird enough, but is the sentence saying the dog is green too, or just that there is also a regular dog? To be clear, we need to ask: "Do you mean a green cat and a regular dog, or a green cat and a green dog, because what you said sounds like green may or may not describe dog?" What we are asking is, "Did you mean \(ab + c\) or \(ab + ac,\) because what you said sounds like it could be \(a(b + c)\) or \(ab + c.\)