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Difficulty: HardSimplifying Expressions and Combining Like Terms

When the expression 3m(m−2n)2−(2m−n)(m2−3mn+2n2)−4n(m2−mn)3m(m - 2n)^2 - (2m - n)(m^2 - 3mn + 2n^2) - 4n(m^2 - mn) is fully simplified by combining like terms, what is the coefficient of the m2nm^2n term?

  1. A
    −23-23
  2. B
    −15-15
  3. −9-9Answer
  4. D
    −3-3
  5. E
    33

Answer

−9-9
The correct answer is −9-9. This is found by carefully expanding each term and distributing the negative signs: the first term expands to 3m3−12m2n+12mn23m^3 - 12m^2n + 12mn^2; the second term, when subtracted, becomes −2m3+7m2n−7mn2+2n3-2m^3 + 7m^2n - 7mn^2 + 2n^3; and the third term simplifies to −4m2n+4mn2-4m^2n + 4mn^2. Summing the coefficients of the m2nm^2n terms gives −12+7−4=−9-12 + 7 - 4 = -9.

Step-by-Step Solution

1
Expand the first term of the expression, 3m(m−2n)23m(m - 2n)^2.
First, square the binomial: (m−2n)2=m2−4mn+4n2(m - 2n)^2 = m^2 - 4mn + 4n^2. Next, distribute 3m3m to get 3m3−12m2n+12mn23m^3 - 12m^2n + 12mn^2.
To remove parentheses from the first term before combining like terms.
2
Expand the product in the second term, (2m−n)(m2−3mn+2n2)(2m - n)(m^2 - 3mn + 2n^2).
2m(m2−3mn+2n2)−n(m2−3mn+2n2)=2m3−6m2n+4mn2−m2n+3mn2−2n32m(m^2 - 3mn + 2n^2) - n(m^2 - 3mn + 2n^2) = 2m^3 - 6m^2n + 4mn^2 - m^2n + 3mn^2 - 2n^3. Combining like terms within this product yields 2m3−7m2n+7mn2−2n32m^3 - 7m^2n + 7mn^2 - 2n^3.
To expand the binomial-trinomial product before applying the subtraction.
3
Subtract the expanded second term and expand the third term, −4n(m2−mn)-4n(m^2 - mn).
Subtracting the second term gives −2m3+7m2n−7mn2+2n3-2m^3 + 7m^2n - 7mn^2 + 2n^3. Distributing the negative sign in the third term gives −4m2n+4mn2-4m^2n + 4mn^2.
To distribute the negative signs across the remaining parenthetical expressions.
4
Identify and combine all the m2nm^2n terms to find the final coefficient.
The m2nm^2n terms are −12m2n-12m^2n (from the first term), +7m2n+7m^2n (from the subtracted second term), and −4m2n-4m^2n (from the third term). Combining these yields (−12+7−4)m2n=−9m2n(-12 + 7 - 4)m^2n = -9m^2n.
To determine the final coefficient of the m2nm^2n term.

Key Concept

Simplifying Expressions and Combining Like Terms
Estimated Time:2m 0s
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Topics

Simplifying Expressions and Combining Like Terms

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