For all real numbers p and q, which of the following is equivalent to the expression 2p^2(… — ACT

For all real numbers $p$ and $q$ , which of the following is equivalent to the expression $2p^2(p - 3q) - 3q(p^2 - q^2) - (2p^3 - 5p^2q)$ ?

-4p^2q + 3q^3

The correct answer is obtained by distributing all factors and negative signs, then combining the coefficients of the like terms. This yields

-4p^2q + 3q^3

Distribute the term

2p^2

into the first parenthesis,

-3q

into the second parenthesis, and the negative sign into the third parenthesis.

2p^2(p - 3q) = 2p^3 - 6p^2q

-3q(p^2 - q^2) = -3p^2q + 3q^3

-(2p^3 - 5p^2q) = -2p^3 + 5p^2q

Expanding the terms removes the parentheses and prepares the expression for combining like terms.

Combine the expanded parts into a single expression and group the like terms together.

(2p^3 - 2p^3) + (-6p^2q - 3p^2q + 5p^2q) + 3q^3

Grouping terms with identical variable parts makes combining coefficients straightforward.

Simplify the coefficients for each group of like terms.

0p^3 + (-6 - 3 + 5)p^2q + 3q^3 = -4p^2q + 3q^3

Simplifying the combined coefficients yields the final, simplified expression.

Simplifying Expressions and Combining Like Terms

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