mrwells2 wrote:

When reverse factoring a^6 - b^6 to (a^3 - b^3) (a^3 + b^3) how do you come to the conclusion that a^3 must equal either b^3 or -b^3 ? Is this a rule that can be applied when factoring any equation? Or just applicable because the equation is = 0 ?

the equation equals to zero whenever that happen... it's like having (x-3)(x+2)=0 the equation yields zero when x=3 [(3-3)(3+2)=0] or when x=-2 (plug in) here we are doing the same thing but with letters. In particular whenever \(a^3=|b^3|\) then our equation will hold true because the result will be zero.

mrwells2 wrote:

Does Statement #2 verify the assumption that a^3 = -b^3 ?

statement two says that a and b have opposit sign, refer back to our two solutions and check what happens whenever a and b have opposite sign (odd exponents don't hide the sign).

If \(a^3=-b^3 ------> a^3+b^3=0 ------> (a^3-b^3)(0)=0\) the equation is valid.

mrwells2 wrote:

Based on statement two you can infer that a is positive and that b is negative.

Therefore (a^3 -(-b)^3) = positive number or 2(a)^3.

And that (a^3 + (-b)^3) = 0 ?

it doesn't necessarely have to be b the negative one. ab<0 case1: a=-ve b=+ve case2: a=+ve b=-ve (and viceversa)

ab<0 tells us that the variables have opposite sign and that they are different from zero.

Hope it shoves some haze away. Let me know.

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