Distributive property vs everything else

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swelloo

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If you can use the distributive property always when multiplying polynomials, besides it being faster what uses do other ways such as FOIL, or Binomial square pattern, or Product of conjugate pairs pattern? Why use or memorize any of these patterns? Or do these patterns have other uses in other fields?
 
swelloo said:
If you can use the distributive property always when multiplying polynomials, besides it being faster what uses do other ways such as FOIL, or Binomial square pattern, or Product of conjugate pairs pattern? Why use or memorize any of these patterns? Or do these patterns have other uses in other fields?
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As far as I know FOIL and "the distributive property" are identical. The other two - Binomial square pattern, or Product of conjugate pairs pattern - I do not know (what are those?)
 
I would guess that the binomial square pattern is \(\displaystyle (a+b)^2=a^2+2ab+b^2\)
and the product of conjugate pairs pattern is \(\displaystyle (a+b)(a-b)=a^2-b^2\)
although I know them as perfect square and difference of two squares.
 
swelloo said:
If you can use the distributive property always when multiplying polynomials, besides it being faster what uses do other ways such as FOIL, or Binomial square pattern, or Product of conjugate pairs pattern? Why use or memorize any of these patterns? Or do these patterns have other uses in other fields?
Click to expand...
The distributive law or FOIL can be used always for expanding, ie multiplying brackets. The other methods you mention are particularly useful when factorising.
 
I can square 21 by knowing the binomial square pattern. I think of 21 as 20+1. Now (20+1)^2 = 20^2 + 2*20*1 + 1^2 = 400 + 40 + 1 = 441. I can do that in my head and so can you.

25^2 = (20 + 5)^2 = 20^2 + 2*20*5 + 5^2 = 400 + 200 + 25 = 625 all in my head since I know the formula.
 
25^2 - 24^2 is trivial if you know the formula. 25^2 - 24^2 = (25+24)(25-24) = 49*1 = 49.
25^2 - 15^2 = (25+15)(25-15) = (40)(10) = 400. All in my head, just because I know the formula.
 
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