The sum or difference of two cubes can be factored into a product of a binomial times a trinomial.
That is,
a3 - b3 = (a - b)(a2 + ab + b2)
a3 + b3 = (a + b)(a2 + ab + b2)
When we have an expression like a3 - b3 or a3 + b3, we can write it as product of a binomial and a trinomial.
In sum of two cubes, the binomial factor on the right side of the equation has a middle sign that is positive. And also, the middle sign of the trinomial factor is always opposite the middle sign of the given problem. Therefore, it is negative.
In difference of two cubes, the binomial factor on the right side of the equation has a middle sign that is negative. And also, the middle sign of the trinomial factor is always opposite the middle sign of the given problem. Therefore, it is positive.
Example 1 :
Factor :
m3 + 8
Solution :
m3 + 8 = m3 + 23
= m3 + 23
= (m + 2)(m2 - 2m + 22)
= (m + 2)(m2 - 2m + 4)
Example 2 :
Factor :
a3 - 125
Solution :
a3 - 125 = a3 - 53
= a3 - 53
= (a - 5)(a2 + 5a + 52)
= (a - 5)(a2 + 5a + 25)
Example 3 :
Factor :
x3 + 8y3
Solution :
x3 + 8y3 = x3 + 23y3
= x3 + (2y)3
= (x + 2y)[x2 - (x)(2y) + (2y)2]
= (x + 2y)(x2 - 2xy + 4y2)
Example 4 :
Factor :
8x3 - 125y3
Solution :
8x3 - 125y3 = 23x3 - 53y3
= (2x)3 - (5y)3
= (2x - 5y)[(2x)2 + (2x)(5y) + (5y)2]
= (2x - 5y)(4x2 + 10xy + 25y2)
Example 5 :
Factor :
27x3 + 64y3
Solution :
27x3 + 64y3 = 33x3 + 43y3
= (3x)3 + (4y)3
= (3x + 4y)[(3x)2 - (3x)(4y) + (4y)2]
= (3x + 4y)(9x2 - 12xy + 16y2)
Example 6 :
Factor :
2m3 - 54n3
Solution :
2m3 - 54n3 = 2(m3 - 27n3)
= 2(m3 - 33n3)
= 2[m3 - (3n)3]
= 2(m - 3n)[m2 + (m)(3n) + (3n)2]
= 2(m - 3n)(m2 + 3mn + 9n2)
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