Difference between Two CubesDate: 11/09/2001 at 14:13:06 From: Robert Sullivan Subject: Difference between two cubes To Dr.Math, Many people know the equation: a^2 - b^2 = (a+b) * (a-b) But what is another equation for a^3 - b^3 I know that a^3 - b^3 = (a*a*a)-(b*b*b), but I'm not an expert on cancelling and transforming equations. Is it possible? Robert Sullivan Date: 11/09/2001 at 15:10:50 From: Doctor Peterson Subject: Re: Difference between two cubes Hi, Robert. Yes, you can factor a^3 - b^3. I'll show you a way to do this. Looking at a^2 - b^2, you can see that it will be zero if a = b. If you have seen anything about solving quadratic equations by factoring, you will recognize that this means that a^2 - b^2 can be factored, with one of the factors being (a-b), which is zero when a = b. The same thing happens with a^3 - b^3; this is also zero when a = b, so it can be factored as (a-b) times something. You can find the other factor by dividing: ________a^2___+_ab____+_b^2_ a - b ) a^3 - b^3 a^3 - a^2 b ----------- a^2 b a^b b - a b^2 ------------- a b^2 - b^3 a b^2 - b^3 ----------- 0 (This is long division of polynomials, which works much like long division of numbers.) This tells us that a^3 - b^3 = (a - b)(a^2 + ab + b^2) I think that is what you were looking for. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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