Question

what is the difference of the polynomials?\\((-2x^{3}y^{2} + 4x^{2}y^{3} - 3xy^{4}) - (6x^{4}y - 5x^{2}y^{3} - y^{5})\\)\\(\circ\\) \\(-6x^{4}y - 2x^{3}y^{2} + 9x^{2}y^{3} - 3xy^{4} + y^{5}\\)\\(\circ\\) \\(-6x^{4}y - 2x^{3}y^{2} - x^{2}y^{3} - 3xy^{4} - y^{5}\\)\\(\circ\\) \\(-6x^{4}y + 3x^{3}y^{2} + 4x^{2}y^{3} - 3xy^{4} + y^{5}\\)\\(\circ\\) \\(-6x^{4}y - 7x^{3}y^{2} + 4x^{2}y^{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 \(-6x^{4}y\) (from the distributed second polynomial, since the first polynomial has no \(x^{4}y\) term).
• For the \(x^{3}y^{2}\) term: There is only \(-2x^{3}y^{2}\) (from the first polynomial, since the second polynomial has no \(x^{3}y^{2}\) term).
• For the \(x^{2}y^{3}\) term: Combine \(4x^{2}y^{3}\) (from the first polynomial) and \(5x^{2}y^{3}\) (from the distributed second polynomial): \(4x^{2}y^{3}+5x^{2}y^{3}=9x^{2}y^{3}\).
• For the \(xy^{4}\) term: There is only \(-3xy^{4}\) (from the first polynomial, since the second polynomial has no \(xy^{4}\) term).
• For the \(y^{5}\) term: There is only \(y^{5}\) (from the distributed second polynomial, since the first polynomial has no \(y^{5}\) term).

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

Answer:

\(-6x^{4}y - 2x^{3}y^{2}+9x^{2}y^{3}-3xy^{4}+y^{5}\) (the first option: \(\boldsymbol{-6x^{4}y - 2x^{3}y^{2}+9x^{2}y^{3}-3xy^{4}+y^{5}}\))