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Difference and sum of cubes

Every expression in the form of a³+b³ or a³-b³ may be factored, or decomposed, into a product of factors in the following way:

a³+b³ = (a+b)(a²-ab+b²)

or

a³-b³ = (a-b)(a²+ab+b²)

Example 1

x³-8 = x³-2³ = (x-2)(x²+2x+2²) = (x-2)(x²+2x+4)

Example 2

x³+8 = x³+2³ = (x+2)(x²-2x+2²) = (x+2)(x²-2x+4)

Example 3

(x+y)³-z³ = [(x+y)-z][(x+y)²+(x+y)z+z²]

Example 4

x⁶-y⁶ = (x²)³-(y²)³ = (x²-y²)[(x²)²+x²y²+(y²)²]
= (x²-y²)(x⁴+x²y²+y⁴) = (x+y)(x-y)(x⁴+x²y²+y⁴)

Example 5

x³+2 = x³ + (3 2)³ = (x+32)(x² -³2x +(³2)²)

Before proceeding to the exercises, ensure that you have a pencil and some paper handy.


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