(1) Factor the difference of two squares:

`a^2-b^2=(a+b)(a-b)`

Note that when you check by multiplication, the terms involving ab cancel.

(2) Factor a perfect square trinomial:

`a^2+2ab+b^2=(a+b)(a+b)=(a+b)^2`

`a^2-2ab+b^2=(a-b)(a-b)=(a-b)^2`

(3) Factor the sum/difference of two cubes:

`a^3-b^3=(a-b)(a^2+ab+b^2)`

`a^3+b^3=(a+b)(a^2-ab+b^2)`

You would have to answer the question about sense making for yourself. To...

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(1) Factor the difference of two squares:

`a^2-b^2=(a+b)(a-b)`

Note that when you check by multiplication, the terms involving ab cancel.

(2) Factor a perfect square trinomial:

`a^2+2ab+b^2=(a+b)(a+b)=(a+b)^2`

`a^2-2ab+b^2=(a-b)(a-b)=(a-b)^2`

(3) Factor the sum/difference of two cubes:

`a^3-b^3=(a-b)(a^2+ab+b^2)`

`a^3+b^3=(a+b)(a^2-ab+b^2)`

You would have to answer the question about sense making for yourself. To me, the perfect square trinomial is easiest to see because I know a geometric interpretation -- draw a square whose sides are a+b -- the square can be partitioned into a square of side a, a square of side b, and 2 rectangles of sides a,b.