Find P if (18m^2 n - 2m n^2 + 15mn) - P = 6m^2 n - 5m n^2 - 12mn.

Step-by-step solution

1. 1

   Rearrange for $P$, noting it is the subtrahend. From $A-P=B$ it follows that $P=A-B$, not $B-A$:

   $P=\left(18m^2n-2mn^2+15mn\right)-\left(6m^2n-5mn^2-12mn\right)$

   Getting this direction backwards would produce the negative of the correct answer, which is by far the most common mistake in this problem type.

2. 2

   Distribute the minus sign over the whole second polynomial. All three signs flip:

   $-\left(6m^2n-5mn^2-12mn\right)=-6m^2n+5mn^2+12mn$

3. 3

   Combine like terms by type. The three distinct term types are $m^2n$, $mn^2$ and $mn$:

   $m^2n:\;18-6=12,\qquad mn^2:\;-2+5=3,\qquad mn:\;15+12=27$

4. 4

   State the result.

   $P=12m^2n+3mn^2+27mn=3mn\left(4m+n+9\right)$

   The factored form comes from the common factor $3mn$ shared by all three terms.

5. 5

   Verify numerically. Take $m=1$, $n=2$. The first polynomial is $18(2)-2(4)+15(2)=36-8+30=58$, and

   $P=12(1)(2)+3(1)(4)+27(2)=24+12+54=90$

   (the factored form agrees: $3mn(4m+n+9)=3\cdot 2\cdot 15=90$). Then the left side is $58-90=-32$, and the right side is $6(2)-5(4)-12(2)=12-20-24=-32\;\checkmark$.