FACTORING SUM AND DIFFERENCE OF CUBES

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. 

Sum of Two Cubes

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.

Difference of Two Cubes

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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