Commutativity and associativity rearrange within a single operation. Distribution is the only property that lets multiplication and addition talk to each other, which makes it the licence behind expanding, factoring and very nearly every rearrangement you will make.
Core ideas
The difficulty with this law is not applying it. It is knowing where it stops, because several false rules have exactly the same shape.
Nothing proves a(b + c) = ab + ac from the other properties — it is one of the assumed properties of the real numbers, stated in “Properties of the Real Numbers”. That is worth knowing, because it means the law is exactly as broad as it is stated and not one case broader.
Every term, not the first one. When the factor is an invisible −1, as in −(x − 7), this is the single most common error in school algebra and it produces answers that look entirely reasonable.
Squaring does not, taking a root does not, and a sum in a denominator does not. Each of those has the shape of the law and none of them is the law; one numerical test settles each in seconds.
The material
Where it comes from, how the minus sign uses it, and four things that resemble it and are false.
For all real numbers a, b and c, a(b + c) = ab + ac.
Multiplication is commutative, so the same statement holds with the factor on the other side: (b + c)a = ba + ca.
One picture, for anyone who wants the law to feel inevitable rather than assumed: a rectangle 3 units high and x + 4 units wide can be cut into a 3 by x piece and a 3 by 4 piece. Its area is 3(x + 4) and it is also 3x + 12. The picture is a reason to believe the law, not a proof of it — it says nothing about negative or fractional lengths, which the law covers and rectangles do not.
The law is stated for addition, and that is enough, because b − c = b + (−c). Distributing over a subtraction is distributing over the sum of b and −c:
Line three is the one to study. A minus sign in front of a bracket is a factor of −1, and it multiplies every term inside. −(x − 7) is −x + 7; it is not −x − 7. Nothing here is being remembered — each sign is computed, and (−1)(−7) = +7 is a fact from “Properties of the Real Numbers”.
The habit worth forming is to write the −1 in. It costs a symbol and converts a sign question into a multiplication, which you can do.
The law places no limit on how many terms are inside the bracket, because it can be applied repeatedly:
The second line is the law used twice: once to split (x + 2) off, and once inside each piece. There is no separate rule for multiplying two brackets; there is this law, applied again. Polynomials and Factoring does this at length and gives it a name.
The last one deserves its own sentence, because division is where the law is genuinely one-sided. (x + 6)/2 does split, giving x/2 + 3, and the reason is that dividing by 2 is multiplying by 1/2, so the law applies with a = 1/2. But 12/(x + 6) is 12 times 1/(x + 6), and 1/(x + 6) is not 1/x + 1/6. A sum you are dividing splits; a sum you are dividing by does not.
ab + ac = a(b + c) is the same statement written backwards, and read that way it is called factoring. 6x + 15 is 3(2x + 5), because both terms share a factor of 3. Polynomials and Factoring is largely the study of how to find the a when nobody tells you what it is.
Interactive
Every term against every term
Method
The first two steps exist entirely to make the third one arithmetic rather than memory.
If the bracket has nothing but a minus in front of it, write (−1). An invisible factor is the one that gets applied to only half the bracket.
x − 7 becomes x + (−7). Now every term inside carries its own sign and there is no subtraction left to mishandle.
One term at a time, left to right, writing each product down before moving on. Count the products against the number of terms in the bracket; they must match.
Substitute a convenient number, x = 1 usually, into the original and into your answer. Disagreement proves you are wrong; agreement is not a proof that you are right, but it catches nearly every sign error.
Each line writes the outside factor explicitly, including where it is an invisible −1.
In lines 2 and 3 the sign that flipped is on the second term, and it flipped because a negative times a negative is positive. Writing −2x − 10 or −x − 7 is what happens when the outside factor is applied to the first term and then merely copied onto the second.
The checks test one value of x each. That is enough to expose a wrong sign and not enough to establish that two expressions agree everywhere — what establishes that is the law itself, which is why the law is stated for all real numbers and the check is called a check.
Fade two of the products for −3(2x − 5): write (−3)(2x) and leave the second product blank before finishing. The expansion is −6x + 15. −6x − 15 kept the inner sign instead of multiplying it; −6x − 5 also failed to send the outside factor to every term.
Expand −4(3x − 2y + 5) and (2x − 3)(x + 4), showing one product for every grid cell. Then use a numerical counterexample to explain why 12/(x + 6) cannot split although (x + 6)/2 can. That shape distinction is what keeps Rational and Radical Expressions from becoming guesswork.
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