Algebra · Algebraic manipulation
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Expanding brackets
Why a multiplier reaches every term inside a bracket, why double brackets need every-with-every (and where the middle term of a square comes from), and how three brackets fall to the same rule two at a time.
Algebra · Algebraic manipulation
Expanding brackets
Why a multiplier reaches every term inside a bracket, why double brackets need every-with-every (and where the middle term of a square comes from), and how three brackets fall to the same rule two at a time.
Why it works
A bracket is a package; multiplying the package multiplies everything in it. means five copies of the whole of , so it must be — five copies of AND five copies of . Stopping halfway () means four of your five packages lost their . Picture a rectangle tall and wide: its area splits into a piece and a piece.A minus in front distributes its sign to every term. : the multiplies the too, and negative × negative is positive. So — watch the .
Double brackets: EVERY term meets EVERY term. is a rectangle with sides split and — four sub-rectangles, four products:
The two middle products then collect. A grid keeps all four honest.
Squaring a bracket is a double bracket — the middle term is not optional. . The tempting shortcut squares the two terms and skips their two cross products ( twice). Test : , but — the shortcut is wrong every time.
Three brackets: same rule, two at a time. Expand any pair first, then multiply the result by the third bracket:
Six products in the second step — a grid with a three-term side keeps count.