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Factoring *a*^{3} - *b*^{3}

An expression of the form *a*^{3} - *b*^{3} is called a difference of
cubes. The factored form of *a*^{3} - *b*^{3} is (*a* - *b*)(*a*^{2} + *ab* + *b*^{2}):

(*a* - *b*)(*a*^{2} + *ab* + *b*^{2}) = *a*^{3} - *a*^{2}*b* + *a*^{2}*b* - *ab*^{2} + *ab*^{2} - *b*^{3} = *a*^{3} - *b*^{3}

For example, the factored form of

27*x*^{3} - 8 (

*a* = 3*x*, *b* = 2) is

(3*x* - 2)(9*x*^{2} + 6*x* + 4).

Similarly, the factored form of

125*x*^{3} -27*y*^{3} (

*a* = 5*x*, *b* = 3*y*) is

(5*x* - 3*y*)(25*x*^{2} +15*xy* + 9*y*^{2}).

To factor a difference of cubes, find *a* and *b* and plug them into (*a* - *b*)(*a*^{2} + *ab* + *b*^{2}).

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Factoring *a*^{3} + *b*^{3}

An expression of the form *a*^{3} + *b*^{3} is called a sum of cubes. The
factored form of *a*^{3} + *b*^{3} is (*a* + *b*)(*a*^{2} - *ab* + *b*^{2}):

(*a* + *b*)(*a*^{2} - *ab* + *b*^{2}) = *a*^{3} + *a*^{2}*b* - *a*^{2}*b* - *ab*^{2} + *ab*^{2} + *b*^{3} = *a*^{3} - *b*^{3}.

For example, the factored form of

64*x*^{3} + 125 (

*a* = 4*x*, *b* = 5) is

(4*x* + 5)(16*x*^{2} - 20*x* + 25).

Similarly, the factored form of

343*x*^{3} + *y*^{3} (

*a* = 7*x*, *b* = *y*) is

(7*x* + *y*)(49*x*^{2} -7*xy* + *y*^{2}).

To factor a sum of cubes, find *a* and *b* and plug them into (*a* + *b*)(*a*^{2} - *ab* + *b*^{2}).

You can remember these two factored forms by remembering that the sign
in the binomial is always the same as the sign in the original
expression, the first sign in the trinomial is the opposite of the sign
in the original expression, and the second sign in the trinomial is
always a plus sign.

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Factoring *ax*^{3} + *bx*^{2} + *cx* + *d*

*ax*^{3} + *bx*^{2} + *cx* + *d* can be easily factored if
=
First, group the terms: (*ax*^{3} + *bx*^{2}) + (*cx* + *d* ).
Next, factor *x*^{2} out of the first group of terms:
*x*^{2}(*ax* + *b*) + (*cx* + *d* ). Factor the constants out of both groups.
This should leave an expression of the form *d*_{1}*x*^{2}(*ex* + *f* )+ *d*_{2}(*ex* + *f* ). We can add these two terms by adding their "coefficients": (*d*_{1}*x*^{2} + *d*_{2})(*ex* + *f* ).