Factoring a3 - b3
An expression of the form a3 - b3 is called a difference of
cubes. The factored form of a3 - b3 is (a - b)(a2 + ab + b2):
(a - b)(a2 + ab + b2) = a3 - a2b + a2b - ab2 + ab2 - b3 = a3 - b3
For example, the factored form of
27x3 - 8 (
a = 3x, b = 2) is
(3x - 2)(9x2 + 6x + 4).
Similarly, the factored form of
125x3 -27y3 (
a = 5x, b = 3y) is
(5x - 3y)(25x2 +15xy + 9y2).
To factor a difference of cubes, find a and b and plug them into (a - b)(a2 + ab + b2).
Factoring a3 + b3
An expression of the form a3 + b3 is called a sum of cubes. The
factored form of a3 + b3 is (a + b)(a2 - ab + b2):
(a + b)(a2 - ab + b2) = a3 + a2b - a2b - ab2 + ab2 + b3 = a3 - b3.
For example, the factored form of
64x3 + 125 (
a = 4x, b = 5) is
(4x + 5)(16x2 - 20x + 25).
Similarly, the factored form of
343x3 + y3 (
a = 7x, b = y) is
(7x + y)(49x2 -7xy + y2).
To factor a sum of cubes, find a and b and plug them into (a + b)(a2 - ab + b2).
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.
Factoring ax3 + bx2 + cx + d
ax3 + bx2 + cx + d can be easily factored if
= 
First, group the terms: (ax3 + bx2) + (cx + d ).
Next, factor x2 out of the first group of terms:
x2(ax + b) + (cx + d ). Factor the constants out of both groups.
This should leave an expression of the form d1x2(ex + f )+ d2(ex + f ). We can add these two terms by adding their "coefficients": (d1x2 + d2)(ex + f ).
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