Pair each unsimplified algebraic expression with its correctly simplified polynomial equiv… — Armed Services Vocational Aptitude Battery (ASVAB)

Pair each unsimplified algebraic expression with its correctly simplified polynomial equivalent in standard form.

$3p(2p - q) - 2(p^2 + 4pq)$ $4p^2 - 11pq$
$-4(p^2 - 2pq) + 2p(p + 3q)$ $-2p^2 + 14pq$
$p(5p + 2q) - (p^2 - 3pq)$ $4p^2 + 5pq$
$2(3p^2 - pq) - p(8p - 5q)$ $-2p^2 + 3pq$

Answer

The expressions simplify as follows: the first expression simplifies to 4p² - 11pq; the second to -2p² + 14pq; the third to 4p² + 5pq; and the fourth to -2p² + 3pq.

Each expression is correctly simplified by first distributing the outer multipliers to every term inside their respective parentheses, paying close attention to negative signs. Once expanded, like terms (terms containing p² and terms containing pq) are combined by adding or subtracting their coefficients.

Step-by-Step Solution

Simplify the first expression: 3p(2p - q) - 2(p² + 4pq)

6p² - 3pq - 2p² - 8pq = 4p² - 11pq

Distribution of the outside terms into the parentheses, followed by grouping like terms.

Simplify the second expression: -4(p² - 2pq) + 2p(p + 3q)

-4p² + 8pq + 2p² + 6pq = -2p² + 14pq

Carefully distribute the negative constant (-4), changing the inner sign to positive 8pq, then combine like terms.

Simplify the third expression: p(5p + 2q) - (p² - 3pq)

5p² + 2pq - p² + 3pq = 4p² + 5pq

Distribute the variable p and distribute the negative sign to flip the inner signs, then collect like terms.

Simplify the fourth expression: 2(3p² - pq) - p(8p - 5q)

6p² - 2pq - 8p² + 5pq = -2p² + 3pq

Distribute the 2 and the -p, ensuring the multiplication of -p and -5q results in +5pq, then combine.

Key Concept

Simplifying Algebraic Expressions

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