multiplication in algebra

Multiplication in Algebra

Consider a Rectangle, with sides (a+b) and (c+d)

The area of the rectangle will then be (a+b)x(c+d)

Now let us draw a rectangle and see what that means

The whole area is made up of 4 small rectangles, whose areas are ac, ad, bc, bd respectively.

Thus, (a+b)x(c+d) = ac + ad + bc + bd

Also, ac +ad = a x (d + c)

And bc + bd = b x (d + c)

Therefore,  the total area can also be written as, thus:

(a+b)x(c+d) = ac + ad + bc + bd = a x (d + c) + b x (d + c)

Please remember:  

 (a+b)x(c+d) =  a x (d + c) + b x (d + c)

also, (a+b)x(c+d) = c x (a + b) + d x (a + b)  (you may like to prove this yourself)