Which of the following is equivalent to the expression 2a(3a - b)^2 - 3b(2a^2 - ab) - a^2(… — ACT

Which of the following is equivalent to the expression $2a(3a - b)^2 - 3b(2a^2 - ab) - a^2(18a - 15b)$ for all real values of $a$ and $b$ ?

The simplified expression is

-3a^2b + 5ab^2

The correct expression is

-3a^2b + 5ab^2

. Expanding the three parts yields

18a^3 - 12a^2b + 2ab^2

-6a^2b + 3ab^2

, and

-18a^3 + 15a^2b

. Summing these parts together cancels out the cubic

a^3

terms (

18a^3 - 18a^3 = 0

), combines the

a^2b

terms (

-12a^2b - 6a^2b + 15a^2b = -3a^2b

), and combines the

ab^2

terms (

2ab^2 + 3ab^2 = 5ab^2

Expand the first term

2a(3a - b)^2

18a^3 - 12a^2b + 2ab^2

First expand the squared binomial

(3a - b)^2 = 9a^2 - 6ab + b^2

, then distribute the

2a

to each term.

Expand the second term

-3b(2a^2 - ab)

-6a^2b + 3ab^2

Distribute

-3b

to both terms inside the parentheses, ensuring that the negative sign is applied to both terms.

Expand the third term

-a^2(18a - 15b)

-18a^3 + 15a^2b

Distribute

-a^2

to both terms inside the parentheses, changing the sign of both terms.

Combine the expanded expressions and group like terms

(18a^3 - 18a^3) + (-12a^2b - 6a^2b + 15a^2b) + (2ab^2 + 3ab^2)

Grouping terms with identical variable parts allows them to be added or subtracted.

Perform the final simplification

-3a^2b + 5ab^2

The

a^3

terms cancel out, the

a^2b

terms combine to

-3a^2b

, and the

ab^2

terms combine to

5ab^2

Simplifying algebraic expressions by expanding parentheses, applying exponent rules, and combining like terms.

Question