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Proof 15: ‘Square of binomial’ and ‘a sum times a difference’

A ‘binomial’ is a word for two added terms, such as: $a+b$. The square of a binomial is naturally $(a+b)^2$.

$$(a+b)^2=(a+b)(a+b)=aa+ab+ba+bb=a^2+b^2+2ab\quad_\blacksquare$$

A small mnemonic rhyme would be: ‘The binomial square is the first term squared plus the second term squared plus the product doubled’. Note that if one of the numbers is negative, the $2ab$ term will automatically get a negative sign: $(a-b)^2=a^2+b^2-2ab$. So, the signs will take care of themselves.

‘A sum times a difference’ means: $(a+b)(a-b)$.

$$(a+b)(a-b)=aa+a(-b)+ba+b(-b)=a^2-\cancel{ab}+\cancel{ab}-b^2=a^2-b^2 \quad_\blacksquare $$

A mnemonic rhyme would be: ‘A sum times a difference is the first term squared minus the second term squared’.

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