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The Distributive Property: Expanding and Factoring Out

One rule, two directions. Multiply across a parenthesis to expand, or pull a common factor out to compress. Meet the distributive property and learn to run it both ways.

The rule itself

The distributive property says a(b + c) = ab + ac: a factor outside a parenthesis multiplies every term inside. The most common mistake is forgetting the “every” — distributing to the first term and dropping the rest. Read left to right, it is expanding; read right to left, it is factoring out a common factor.

Expanding (left to right):
   3(2x + 5) = 3*2x + 3*5 = 6x + 15

   -2(x - 4) = -2*x + (-2)*(-4) = -2x + 8
        (watch the sign: minus times minus is plus)

Factoring out (right to left):
   6x + 15 = 3*(2x) + 3*(5) = 3(2x + 5)
        the common factor 3 comes back outside
The same property used both ways, with the sign on -2 handled carefully.

Expanding cleanly

  1. Multiply the outside factor by the first inside term, then by the second, and so on — one arrow to each term.
  2. Apply the sign rules at each multiplication; a negative outside factor flips every inside sign.
  3. After expanding, sweep up any like terms that now sit side by side.
Expand and simplify:  2(3x - 1) + 4(x + 2)

   2(3x - 1) = 6x - 2
   4(x + 2)  = 4x + 8

   Add:  6x - 2 + 4x + 8
   Combine like terms:  (6x + 4x) + (-2 + 8) = 10x + 6
Distribute each parenthesis, then combine like terms to finish.

Factoring out a common factor

Going the other way, you look for the largest factor every term shares — the common monomial factor — and pull it out front. To factor 12x^2 + 8x, notice both terms share 4 and an x, so the common factor is 4x, leaving 4x(3x + 2). You can always check by expanding again.