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Difficulty: MediumSimplifying Expressions and Combining Like Terms

When the expression 2a(a2−3ab)−3b(a2+2b2)−(a3−5ab2)2a(a^2 - 3ab) - 3b(a^2 + 2b^2) - (a^3 - 5ab^2) is simplified by combining like terms, what is the coefficient of the a2ba^2b term?

Answer: -9

Answer

The coefficient of the a2ba^2b term is −9-9.
Expanding the entire expression yields 2a3−6a2b−3a2b−6b3−a3+5ab22a^3 - 6a^2b - 3a^2b - 6b^3 - a^3 + 5ab^2. Combining the a2ba^2b terms gives (−6−3)a2b=−9a2b(-6 - 3)a^2b = -9a^2b. Therefore, the coefficient of the a2ba^2b term is −9-9.

Step-by-Step Solution

1
Distribute 2a2a to the terms inside the first set of parentheses: 2a(a2−3ab)2a(a^2 - 3ab)
2a3−6a2b2a^3 - 6a^2b
To expand the first part of the expression.
2
Distribute −3b-3b to the terms inside the second set of parentheses: −3b(a2+2b2)-3b(a^2 + 2b^2)
−3a2b−6b3-3a^2b - 6b^3
To expand the second part of the expression, ensuring the negative sign is distributed to all terms inside.
3
Distribute the negative sign to the terms inside the third set of parentheses: −(a3−5ab2)-(a^3 - 5ab^2)
−a3+5ab2-a^3 + 5ab^2
To expand the third part of the expression, reversing the sign of each term inside.
4
Identify and combine the like terms for the a2ba^2b variable combination: −6a2b−3a2b-6a^2b - 3a^2b
−9a2b-9a^2b
To simplify the expression by combining terms with the same variable components.

Key Concept

Simplifying expressions by distributing terms and combining like terms
Estimated Time:1m 30s
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Topics

Simplifying Expressions and Combining Like Terms

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