The grouping method is really just the reverse of FOIL (first, outer, inner, last) expansion
(cx+d)=(ac)x^2+(ad+bc)x+bd)
as for the grouping method, when we multiply the first coefficient with the constant... we get
and then the two numbers we're looking for needs to multiply to the product of the first coefficient and the constant (i.e.
), and need to add to the second coefficient (
), hence by trail and error, we arrive at the two numbers, namely
and
reverse foil is as follows:
x^2+(ad+bc)x+bd)
x^2 + (ad)x + bc(x) + bd)
 + b(cx+d))
(cx+d))
as can be seen, for this method to work the second coefficient need to be broken down correctly for reverse FOIL to happen, hence why we first need to find these two numbers.
this is what ice_blockie referred to as the "grouping method"
it may seem complex here, but practical application of this method usually involve equations a lot simpler:
(x+b) = x^2 + (a+b)x + ab)
so as soon as you find out a and b, you can factorise without having to reverse FOIL
^2 = x^2 + (2a)x + a^2)
you can get "a" by just looking at it
^2 = a^2 x^2 + (2ab)x + b^2)
also fairly simple mental arithmetics
other methods that are also useful:
Complete the square (ice blockie question
Remainder theorem (havent seen around)
Synthetic method of polynomial factorisation (totally forgot how that works *looks up in wikipedia*)
so yeah....
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