For all real values of a and b, which of the following is equivalent to the expression 2a(… — ACT

For all real values of $a$ and $b$ , which of the following is equivalent to the expression $2a(a^2 - 3b) - 3(a^3 - 2ab + b^2) - (4b^2 - a^3)$ ?

-7b^2

Distributing the terms yields

2a^3 - 6ab - 3a^3 + 6ab - 3b^2 - 4b^2 + a^3

. Combining the like terms for

a^3

gives

2a^3 - 3a^3 + a^3 = 0

. Combining the

ab

terms gives

-6ab + 6ab = 0

. Combining the

b^2

terms gives

-3b^2 - 4b^2 = -7b^2

. Thus, the simplified expression is

-7b^2

Distribute the factors outside the parentheses to each term inside the parentheses.

The terms expand to:

2a(a^2) - 2a(3b) - 3(a^3) - 3(-2ab) - 3(b^2) - 1(4b^2) - 1(-a^3) = 2a^3 - 6ab - 3a^3 + 6ab - 3b^2 - 4b^2 + a^3

Distribution eliminates parentheses, making it possible to group and combine like terms.

Group like terms together based on their variable parts and powers.

(2a^3 - 3a^3 + a^3) + (-6ab + 6ab) + (-3b^2 - 4b^2)

Grouping like terms makes it easier to perform the arithmetic on the coefficients.

Combine the coefficients for each group of like terms.

0a^3 + 0ab - 7b^2 = -7b^2

Simplifying the coefficient sums yields the final simplified form of the expression.

Simplifying expressions by distributing terms and combining like terms

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