Depth · one concept from first principles · step 2 of 4

Undo the wrong step to recover the polynomial

Someone was supposed to divide a polynomial by $2xy$ , but multiplied it by $2xy$ by mistake and got $32x^3y^2-48x^2y^3$ . What is the result of the correct calculation?

Expressions

Your answer

How to solve

Strategy Work Backwards — The mistaken result is an end state. Name the lost polynomial, undo the multiplication to get it back, and only then carry out the division that was intended.

1STEP 1

Name the lost polynomial

Call the original polynomial P. The mistake says P times 2xy came out as the given result.

P × 2xy = 32x³y²-48x²y³

2STEP 2

Undo the multiplication

Dividing by 2xy undoes multiplying by 2xy. Divide each term separately, subtracting exponents of matching letters.

P = 32x³y²/(2xy)-48x²y³/(2xy) = 16x²y-24xy²

3STEP 3

Do the intended division

Now divide the recovered polynomial by 2xy, again term by term (the distributive property).

(16x²y-24xy²) ÷ 2xy = 8x-12y

Answer

8x-12y

Going from the mistaken result to the right one means dividing by 2xy twice, that is by 4x²y²: 32x³y²/(4x²y²)=8x and 48x²y³/(4x²y²)=12y. Same answer, 8x-12y.

Takeaway

When a step was done the wrong way, undo it first, then do it the right way.

Name the lost polynomial
Undo the multiplication
Do the intended division

Where next?

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▶ Practice — 12 problems

The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Branching State Enumeration, Multiplicative Ratio with Integrality Constraints. Why the shapes repeat →