What is expansion?
Expansion means removing brackets by multiplying. The distributive law says every term inside the bracket is multiplied by the term outside it: a(b + c) = ab + ac. For example, 3(x + 4) = 3x + 12.
Key idea / 要点
Single bracket: a(b + c) = ab + ac. Two brackets: (a + b)(c + d) = ac + ad + bc + bd — multiply every term in the first bracket by every term in the second.
Two brackets
To expand (x + 2)(x + 3), multiply each term: x·x + x·3 + 2·x + 2·3 = x2 + 3x + 2x + 6 = x2 + 5x + 6. Always collect the like terms (here 3x + 2x = 5x) at the end.
Worked example / 例题
Expand (x + 4)2.
(x + 4)2 = (x + 4)(x + 4) = x2 + 4x + 4x + 16 = x2 + 8x + 16.
Special products
Two patterns are worth remembering. The difference of two squares: (a + b)(a − b) = a2 − b2. For example (x + 5)(x − 5) = x2 − 25. And the perfect square: (a + b)2 = a2 + 2ab + b2. Spotting these saves time.
Remember / 记住
- Multiply every term, watching the signs.
- Collect like terms after expanding.
- x · x = x2, not 2x.