Special Products

Algebra 1 · Axiom Academy

Squaring a binomial and multiplying a sum by a difference both collapse into patterns you can read straight off an area model. To square the binomial (a + b) means to multiply it by itself: (a + b)(a + b). Picture a square whose side length is a + b. Splitting that side into an a part and a b part chops the square into four tiles whose areas are the four products. 2. Square of a Difference: (a − b)² Now square a binomial with subtraction: (a − b)² . Geometrically we start from the full a × a square and remove what is NOT inside the smaller (a − b) square — an L-shaped border. Watch which pieces leave and which one we have to add back. 3. Difference of Squares: (a + b)(a − b) Multiply a sum by a difference of the same two terms and something clean happens: the cross terms are equal and opposite, so they cancel . Build the rectangle that is (a + b) wide and (a − b) tall and watch the +ab and -ab pieces wipe each other out. Every special product is just these three rules with a and b filled in: Run them in reverse and you can factor : spot a^2 - b^2 and it splits into (a+b)(a-b) ; spot a^2 + 2ab + b^2 and it came from (a+b)^2 . Three products you can now read straight off the rectangle — and run backward to factor. Scroll up to replay any area model.

This is the written version of the interactive lesson above. See the full Algebra 1 course.