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**HallsofIvy** To multiply together two expressions, you multiply **every** term of the first by **every** term of the last. "FOIL" is just a way of being sure you get all of them and only apply to "two terms times two terms".

For example: to multiply (a+ b)(c+ d) I can use "FOIL": "First" terms, ab; "Outside terms", ad; "Inside" terms, bc; "Last" terms, bd. Adding all of those: ab+ ad+ bc+ bd.

I could have got the same result by using the rule: Multiply the first term in the first expression, a, by each of the terms in the second expression: a(c+ c)= ac+ ad. Then multiply the second term in the first expression by each of the terms in the second expression: b(c+ d)= bc+ bd. Adding those: ac+ ad+ bc+ bd, exactly as above.

If I have 3 terms in, say, the first expression, like (a+ b+ c)(u+ v), I can't use "FOIL" exactly but I can think:

Multiply the first term in the first expression, a, by every term in the second expression- a(u+ v)= au+ av.

Multiply the second term in the first expression, b, by every term in the second expression- b(u+ v)= bu+ bv.

Multiply the third term in the first expression, c, by every term in the second expression- c(u+ v)= cu+ cv.

Now add all of those: (a+ b+ c)(u+ v)= au+ av+ bu+ bv+ cu+ cv.