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Question

Difficulty: MediumSimplifying Algebraic Expressions

Which of the following is equivalent to the algebraic expression below after it has been fully simplified?

4k(k−3m)−(2k2−5km+m2)4k(k - 3m) - (2k^2 - 5km + m^2)
  1. 2k2−7km−m22k^2 - 7km - m^2Answer
  2. B
    2k2−17km+m22k^2 - 17km + m^2
  3. C
    6k2−7km−m26k^2 - 7km - m^2
  4. D
    2k2−8km22k^2 - 8km^2

Answer

The fully simplified expression is 2k2−7km−m22k^2 - 7km - m^2.
The correct expression is derived by first multiplying 4k4k by both terms in (k−3m)(k - 3m) to get 4k2−12km4k^2 - 12km. Next, distributing the negative sign through the trinomial changes the signs to yield −2k2+5km−m2-2k^2 + 5km - m^2. Finally, grouping and combining like terms yields 4k2−2k2=2k24k^2 - 2k^2 = 2k^2 and −12km+5km=−7km-12km + 5km = -7km. The −m2-m^2 term has no like terms to combine with, leaving it unchanged. The final simplified form is 2k2−7km−m22k^2 - 7km - m^2.

Step-by-Step Solution

1
Distribute 4k4k into the binomial (k−3m)(k - 3m).
4k2−12km4k^2 - 12km
Expanding the first part of the expression removes the first set of parentheses.
2
Distribute the negative sign across the trinomial (2k2−5km+m2)(2k^2 - 5km + m^2).
−2k2+5km−m2-2k^2 + 5km - m^2
Subtracting a polynomial requires reversing the sign of every term inside the parentheses.
3
Group and combine all like terms.
(4k2−2k2)+(−12km+5km)−m2=2k2−7km−m2(4k^2 - 2k^2) + (-12km + 5km) - m^2 = 2k^2 - 7km - m^2
Expressions are fully simplified when all terms with identical variable combinations are merged.

Key Concept

Distributing variables and negative signs followed by combining like terms.
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Topics

Simplifying Algebraic Expressions

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