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square of sum∗
pahio†
2013-03-21 19:44:50
The well-known formula for squaring a sum of two numbers or terms is
(a+b)2 = a2 +2ab+b2 .
(1)
It may be derived by multiplying the binomial a+b by itself.
Similarly one can get the squaring formula for a sum of three summands:
(a+b+c)2 = a2 +b2 +c2 +2bc+2ca+2ab
(2)
Its contents may be expressed as the
Rule. The square of a sum is equal to the sum of the squares of all the
summands plus the sum of all the double products of the summands in twos:
!2
X
ai
=
X
i
i
a2i + 2
X
ai aj .
i<j
This is true for any number of summands. The rule may be formulated also
as
(a+b+c + ...)2 = (a)a + (2a+b)b + (2a+2b+c)c + ...
(3)
which in the case of four summands is
(a+b+c+d)2 = (a)a + (2a+b)b + (2a+2b+c)c + (2a+2b+2c+d)d.
(4)
One can use the idea of (3) to find the square root of a polynomial, when one
tries to arrange the polynomial into the form of the right hand side of (3).
∗ hSquareOfSumi
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