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Proof of ${(a+b+c)}^2$ formula in Algebraic Method

Proof

The expansion of a plus b plus c whole square formula can be derived in mathematics by the multiplication of algebraic expressions.

Multiplying Trinomials

The square of the trinomial $a+b+c$ can be expanded by multiplying two same trinomials.

$\implies$ ${(a+b+c)}^2$ $\,=\,$ ${(a+b+c)}$ $\times$ ${(a+b+c)}$

Multiply the second trinomial by the each term of the first trinomial.

$\implies$ ${(a+b+c)}^2$ $\,=\,$ $a \times {(a+b+c)}$ $+$ $b \times {(a+b+c)}$ $+$ $c \times {(a+b+c)}$

Multiply each term of the trinomial by associated multiplying term.

$\implies$ ${(a+b+c)}^2$ $\,=\,$ $a \times a$ $+$ $a \times b$ $+$ $a \times c$ $+$ $b \times a$ $+$ $b \times b$ $+$ $b \times c$ $+$ $c \times a$ $+$ $c \times b$ $+$ $c \times c$

Simplify each term by the product rule of the algebraic terms.

$\implies$ ${(a+b+c)}^2$ $\,=\,$ $a^2$ $+$ $ab$ $+$ $ac$ $+$ $ba$ $+$ $b^2$ $+$ $bc$ $+$ $ca$ $+$ $cb$ $+$ $c^2$

Identifying the Like terms

Write all algebraic terms in appropriate order to simply the expression easily.

$\implies$ ${(a+b+c)}^2$ $\,=\,$ $a^2$ $+$ $b^2$ $+$ $c^2$ $+$ $ab$ $+$ $ac$ $+$ $ba$ $+$ $bc$ $+$ $ca$ $+$ $cb$

$\implies$ ${(a+b+c)}^2$ $\,=\,$ $a^2$ $+$ $b^2$ $+$ $c^2$ $+$ $ab$ $+$ $ba$ $+$ $bc$ $+$ $cb$ $+$ $ca$ $+$ $ac$

The product of any two literals in any order is always equal.

$\implies$ ${(a+b+c)}^2$ $\,=\,$ $a^2$ $+$ $b^2$ $+$ $c^2$ $+$ $ab$ $+$ $ab$ $+$ $bc$ $+$ $bc$ $+$ $ca$ $+$ $ca$

Adding Like terms

There are three types of like terms in the expression and add all of them to obtain the expansion of the square of $a$ plus $b$ plus $c$.

$\,\,\, \therefore \,\,\,\,\,\,$ ${(a+b+c)}^2$ $\,=\,$ $a^2$ $+$ $b^2$ $+$ $c^2$ $+$ $2ab$ $+$ $2bc$ $+$ $2ca$