Question

what is the difference of the polynomials?\\((-2x^3y^2 + 4x^2y^3 - 3xy^4) - (6x^4y - 5x^2y^3 - y^5)\\)\\(\bigcirc -6x^4y - 2x^3y^2 + 9x^2y^3 - 3xy^4 + y^5\\)\\(\bigcirc -6x^4y - 2x^3y^2 - x^2y^3 - 3xy^4 - y^5\\)\\(\bigcirc -6x^4y + 3x^3y^2 + 4x^2y^3 - 3xy^4 + y^5\\)\\(\bigcirc -6x^4y - 7x^3y^2 + 4x^2y^3 - 3xy^4 - y^5\\)

Explanation:

Step1: Distribute the negative sign

To subtract the second polynomial from the first, we distribute the negative sign to each term in the second polynomial:
$$(-2x^{3}y^{2}+4x^{2}y^{3}-3xy^{4}) - 6x^{4}y + 5x^{2}y^{3}+y^{5}$$

Step2: Combine like terms

• For the \(x^{4}y\) term: There is only \(-6x^{4}y\) (since the first polynomial has no \(x^{4}y\) term).
• For the \(x^{3}y^{2}\) term: There is only \(-2x^{3}y^{2}\) (since the second polynomial has no \(x^{3}y^{2}\) term after distribution).
• For the \(x^{2}y^{3}\) term: \(4x^{2}y^{3}+5x^{2}y^{3}=9x^{2}y^{3}\).
• For the \(xy^{4}\) term: There is only \(-3xy^{4}\) (since the second polynomial has no \(xy^{4}\) term after distribution).
• For the \(y^{5}\) term: There is only \(y^{5}\) (since the first polynomial has no \(y^{5}\) term before distribution).

Putting it all together, we get:
$$-6x^{4}y - 2x^{3}y^{2}+9x^{2}y^{3}-3xy^{4}+y^{5}$$

Answer:

-6x⁴y - 2x³y² + 9x²y³ - 3xy⁴ + y⁵ (corresponding to the first option)