What is the difference of the polynomials (−2x³y² + 4x²y³ − 3xy⁴) − (6x⁴y − 5x²y³ − y⁵)?

Step-by-step solution

1. 1

   Distribute the minus sign across the whole second bracket. Subtraction flips the sign of every term inside, including the ones that are already negative:

   $-(6x^4y - 5x^2y^3 - y^5) = -6x^4y + 5x^2y^3 + y^5$

   Flipping only the first term is the usual mistake and produces $-x^2y^3 - y^5$ instead of the correct entries.

2. 2

   Write out the full unbracketed expression.

   $-2x^3y^2 + 4x^2y^3 - 3xy^4 - 6x^4y + 5x^2y^3 + y^5$

3. 3

   Identify which terms are genuinely like terms. In two variables, terms combine only when both exponents match. Comparing the monomials $x^4y$, $x^3y^2$, $x^2y^3$, $xy^4$, $y^5$, the only repeated pattern is $x^2y^3$, which appears twice. All other terms are alone.

4. 4

   Combine the one matching pair.

   $4x^2y^3 + 5x^2y^3 = 9x^2y^3$

   The coefficient becomes $+9$ because the subtraction already turned $-5x^2y^3$ into $+5x^2y^3$.

5. 5

   Order the result by degree in x. Collecting the surviving terms from the highest power of $x$ downwards:

   $-6x^4y - 2x^3y^2 + 9x^2y^3 - 3xy^4 + y^5$

   Every term has total degree $5$, which is a useful consistency check for a homogeneous expression like this one.

6. 6

   Verify with a numerical substitution. Taking $x = 2$, $y = 1$: the original expression is $(-16 + 16 - 6) - (96 - 20 - 1) = -6 - 75 = -81$, and the answer gives $-96 - 16 + 36 - 6 + 1 = -81$. The two agree, confirming all five coefficients.