Difference of Cubes

Q:  Simplify (x3 – 27) / (x-3)

A:  First, to answer this, we must know the formula to factor a “difference of cubes”… Since x3 – 27 = x3 – 33 , this is what we call a difference of cubes (two numbers being raised to the 3rd power, subtracting from each other).  The “difference of cubes” factoring formula is one you just need to memorize… And, it is:

a3b3 = (ab)(a2 + ab + b2)

So, matching above:

x333 = (x3)(x2 + x*3 + 32)

= (x – 3)(x² + 3x + 9)

So, our original problem was: (x3 – 27) / (x-3), and now we can factor the top and plug it in (like we just did):

(x3 – 27) / (x-3) = (x – 3)(x² + 3x + 9)/ (x-3)

And cancel:

(x – 3)(x² + 3x + 9)/ (x-3)

= (x² + 3x + 9)

TADA!

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