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Given only the expression a² – b², it might be difficult to deduce the factors which produce it. One approach, useful in very many other contexts as well, endeavours to find values which make the expression zero. Since a product is zero only if at least one of the factors is zero¹, this suggests factors. Because a² – b² = 0 when a = b, and then a – b is also zero it is potentially a factor. A factor is confirmed and remaining factors isolated by division. The details are omitted, as above it is proven to be a factor by multiplication. Similarly a² – b² = 0 when a = –b, suggesting a + b is possibly a factor, as also proven by the multiplication above.

Division by Zero

Reconsidering the above, it was stated that as a² – b² = 0 when a = b suggests a – b as a factor, which could be confirmed by division. That division would be (a² – b²) ⁄ (a – b), but attempting 0 ⁄ 0 is a widely recognised route to self delusion and madness. As this procedure is widely used in algebra, it will be shown to be entirely justified and valid, and therefore a completely sane maneouvre, as division never actually occurs!

The 'division' of a² – b² by a – b begins by noting the highest powers of a in numerator and denominator, that is a² and a respectively. Then, although it is said that a² is divided by a to give a, this is not quite true, since a may be zero. In fact it is determined that a² = a × a, which does not involve division by zero, but multiplication by a, which is valid even when a = 0. Having found this 'quotient', the product a(a – b) is formed. Subtracting this from a² – b² yields a² – b² – (a² – ab) = ab – b² as a new 'remainder', with which the 'division' continues, having so far established that a² – b² = a(a – b) + (ab – b²). 'Dividing' ab – b² by a – b by noting the highest powers of a in numerator and denominator, that is ab and a, it is determined that ab = a × b, which does not involve division by zero, but multiplication by b. The product b(a – b) is formed and subtracted from ab – b², leaving zero, that is no further remainder. This establishes that a² – b² = a(a – b) + b(a – b) = (a – b)(a + b). The 'division' is valid at every step.

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¹ it is possible that a product of non-zero factors is zero, but only in advanced contexts such as matrix algebra