Factor completely.
(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?
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 )
Yes, your factorization is correct. The expression (m+n)^3 - 27 can be factored completely as (m+n-3)((m+n)^2 + 3(m+n) + 9).
To arrive at this solution, you first substituted a with m+n and b with 3 to get (m+n-3)[(m+n)^2+3(m+n)+3^2]. This is a good approach to simplify expressions.
Next, you expanded (m+n)^2 + 3(m+n) + 9, which gives you m^2 + 2mn + n^2 + 3m + 3n + 9.
Finally, you applied the distributive property to write the factored form as (m+n-3)[(m+n)^2+3(m+n)+3^2], which simplifies to (m+n-3)(m^2 + 2mn + n^2 + 3m + 3n + 9).
Well done!