Match each algebraic expression on the left with its fully simplified equivalent expressio… — ACT

Match each algebraic expression on the left with its fully simplified equivalent expression on the right. All variables represent real numbers.

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

Answer

The first expression

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

matches

2a^3 - a^2b

. The second expression

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

matches

a^2b

. The third expression

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

matches

2a^3

Each expression on the left is simplified by distributing the coefficients and combining the like terms. The first expression simplifies to

2a^3 - a^2b

. The second expression simplifies to

a^2b

. The third expression simplifies to

2a^3

. These match the corresponding simplified expressions on the right.

Step-by-Step Solution

Simplify the first expression

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

2a^3 - a^2b

Multiply

3a

by both terms in the first parentheses to get

3a^3 - 6a^2b

. Distribute the negative sign to both terms in the second parentheses to get

-a^3 + 5a^2b

. Combine the

a^3

terms to get

2a^3

and the

a^2b

terms to get

-a^2b

Simplify the second expression

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

a^2b

Multiply

a^2

by both terms in the first parentheses to get

2a^3 - a^2b

. Multiply

-2a

by both terms in the second parentheses to get

-2a^3 + 2a^2b

. Combine the

a^3

terms to get

0

and the

a^2b

terms to get

a^2b

Simplify the third expression

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

2a^3

Distribute the negative sign to get

-a^3 - 3a^2b

. Multiply

3a^2

by both terms in the second parentheses to get

3a^3 + 3a^2b

. Combine the

a^3

terms to get

2a^3

and the

a^2b

terms to get

0

Key Concept

Simplifying Expressions and Combining Like Terms

Estimated Time:1m 30s

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