Evaluate the following multivariable algebraic expressions. Pair each original expression … — Armed Services Vocational Aptitude Battery (ASVAB)

Evaluate the following multivariable algebraic expressions. Pair each original expression with its simplified form in standard notation.

$-3a(2a - b) - (a - 2b)(2a + b)$ $-8a^2 + 6ab + 2b^2$
$2ab(a - 3b) - a(2ab - 5b^2)$ $-ab^2$
$4(a^2 - ab + b^2) - (2a - b)^2$ $3b^2$
$a^2(2b - 1) - b(a^2 - 2a) + (a - b)^2 - b^2$ $a^2b$

Answer

Each unsimplified expression matches exactly one standard form polynomial upon successful expansion and collection of like terms.

Each pairing is correct because the unsimplified expressions rigidly evaluate to the exact standard forms provided when algebraic expansion and simplification rules are followed without computational errors.

Step-by-Step Solution

Simplify the first expression by expanding and subtracting the grouped terms.

-3a(2a - b) - (a - 2b)(2a + b)

simplifies to

-8a^2 + 6ab + 2b^2

Applying the distributive property and FOIL method accurately, followed by subtracting the terms (which reverses their signs).

Simplify the second expression by applying distribution to both halves.

2ab(a - 3b) - a(2ab - 5b^2)

simplifies to

-ab^2

Distributing the multipliers across the parentheses isolates terms that naturally cancel or combine.

Simplify the third expression by expanding the perfect square.

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

simplifies to

3b^2

The expansion of

(2a - b)^2

creates terms that precisely offset the

a^2

and

ab

terms in the distributed trinomial.

Simplify the fourth expression by expanding all grouped and squared segments.

a^2(2b - 1) - b(a^2 - 2a) + (a - b)^2 - b^2

simplifies to

a^2b

Combining the resulting elements reveals that nearly all terms possess additive inverses, leaving only the combination of

2a^2b - a^2b

Key Concept

Expanding polynomials, applying the distributive property with negative signs, and collecting like variables.

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