Match each algebraic expression on the left with its fully simplified equivalent expressio… — ACT

Match each algebraic expression on the left with its fully simplified equivalent expression on the right for all real values of $m$ and $n$ .

The correct matches are: (1)

m(m^2 - 3mn) - 2n(m^2 - n^2) - (m^3 - mn^2)

matches with

-5m^2n + mn^2 + 2n^3

; (2)

(m - n)^3 - m(m^2 - 3n^2) + n^3

matches with

-3m^2n + 6mn^2

; (3)

3mn(m - n) - 2m(n^2 - mn) - (m^2n - 5mn^2)

matches with

4m^2n

; and (4)

m^2(2m - n) - n(m^2 - n^2) - (m^3 + n^3)

matches with

m^3 - 2m^2n

Each expression is simplified by systematically expanding parenthetical groups and collecting like terms with identical variable powers.

Simplify the expression

m(m^2 - 3mn) - 2n(m^2 - n^2) - (m^3 - mn^2)

-5m^2n + mn^2 + 2n^3

Distribute each term across the parenthetical expressions:

m^3 - 3m^2n - 2m^2n + 2n^3 - m^3 + mn^2

. Group the like terms:

(m^3 - m^3) + (-3m^2n - 2m^2n) + mn^2 + 2n^3

, which simplifies to

-5m^2n + mn^2 + 2n^3

Simplify the expression

(m - n)^3 - m(m^2 - 3n^2) + n^3

-3m^2n + 6mn^2

Use the binomial expansion formula to expand

(m - n)^3 = m^3 - 3m^2n + 3mn^2 - n^3

. Distribute the

-m

term to obtain

-m^3 + 3mn^2

. Sum all the expressions and combine like terms:

(m^3 - m^3) - 3m^2n + (3mn^2 + 3mn^2) + (-n^3 + n^3) = -3m^2n + 6mn^2

Simplify the expression

3mn(m - n) - 2m(n^2 - mn) - (m^2n - 5mn^2)

4m^2n

Expand the terms by distributing the outer coefficients to get

3m^2n - 3mn^2 - 2mn^2 + 2m^2n - m^2n + 5mn^2

. Grouping similar variables yields

(3 + 2 - 1)m^2n + (-3 - 2 + 5)mn^2 = 4m^2n + 0 = 4m^2n

Simplify the expression

m^2(2m - n) - n(m^2 - n^2) - (m^3 + n^3)

m^3 - 2m^2n

Expand the terms by distributing the multiplication:

2m^3 - m^2n - m^2n + n^3 - m^3 - n^3

. Grouping like terms yields

(2m^3 - m^3) + (-m^2n - m^2n) + (n^3 - n^3) = m^3 - 2m^2n

Simplifying multivariable expressions by distributing terms (including negative signs) and combining like terms.

Question