Factor completely.

Question

(m+n)^3-27
a=m+n b=3
(m+n-3)[(m+n)^2+3(m+n)+3^2]
m^3 +3m^2n +3mn^2 +n^3-9
Is this right?

Answer 1

Substitution:

k = m + n

( m + n )³ - 3³ = k³ - 3³

Apply the difference of cubes formula:

a³ - b³ = ( a - b ) ( a² + a b + b² )

k³ - 3³ = ( k - 3 ) ( k² + k ∙ 3 + 3² ) =

( k - 3 ) ( k² + 3 k + 3² )

Return to the initial variables:

( m + n )³ - 3³ = k³ - 3³ =

( k - 3 ) ( k² + 3 k + 3² ) =

( m + n - 3 ) [ ( m + n )² + 3 ∙ ( m + n ) + 3² ]

( m + n )³ - 3³ = ( m + n - 3 ) [ ( m + n )² + 3 ( m + n ) + 3² ]

is correct

Now:

________________________
( m + n )² = m² + 2 m n + n²
_______________________

( m + n - 3 ) ( m² + 2 m n + n² + 3 m + 3 n + 3² ) =

( m + n - 3 ) ( m² + 2 m n + n² + 3 m + 3 n + 9 )

( m + n )³ - 3³ = ( m + n - 3 ) ( m² + 2 m n + n² + 3 m + 3 n + 9 )