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Difficulty: MediumSimplifying Algebraic Expressions

Consider the algebraic expression:

3a(2a−5b)−2b(a+4b)+7ab3a(2a - 5b) - 2b(a + 4b) + 7ab

When completely simplified, what is the resulting polynomial?

  1. 6a2−10ab−8b26a^2 - 10ab - 8b^2Answer
  2. B
    6a2−10ab+8b26a^2 - 10ab + 8b^2
  3. C
    6a2−6ab−8b26a^2 - 6ab - 8b^2
  4. D
    −12ab-12ab

Answer

The simplified polynomial is 6a2−10ab−8b26a^2 - 10ab - 8b^2.
The correct answer is found by first distributing the outer terms to eliminate parentheses, carefully observing negative signs. Distributing 3a3a gives 6a2−15ab6a^2 - 15ab. Distributing −2b-2b gives −2ab−8b2-2ab - 8b^2. Adding the +7ab+7ab at the end produces the expanded expression 6a2−15ab−2ab−8b2+7ab6a^2 - 15ab - 2ab - 8b^2 + 7ab. Combining the like terms (−15ab−2ab+7ab-15ab - 2ab + 7ab) yields −10ab-10ab, resulting in the final expression 6a2−10ab−8b26a^2 - 10ab - 8b^2.

Step-by-Step Solution

1
Distribute 3a3a to the terms inside the first set of parentheses.
3a(2a)−3a(5b)=6a2−15ab3a(2a) - 3a(5b) = 6a^2 - 15ab
Applying the distributive property to eliminate the first set of parentheses.
2
Distribute −2b-2b to the terms inside the second set of parentheses.
−2b(a)−2b(4b)=−2ab−8b2-2b(a) - 2b(4b) = -2ab - 8b^2
Applying the distributive property, ensuring the negative sign is multiplied into both terms.
3
Rewrite the full expression with the expanded terms.
6a2−15ab−2ab−8b2+7ab6a^2 - 15ab - 2ab - 8b^2 + 7ab
Putting all terms together to prepare for combining like terms.
4
Identify and combine the like terms (the abab terms).
−15ab−2ab+7ab=−10ab-15ab - 2ab + 7ab = -10ab
Only terms with the exact same variable combinations can be added or subtracted.
5
Construct the final polynomial.
6a2−10ab−8b26a^2 - 10ab - 8b^2
The a2a^2 and b2b^2 terms cannot be combined with anything else.

Key Concept

Simplifying algebraic expressions using the distributive property and combining like terms.
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Topics

Simplifying Algebraic Expressions

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