which is true about the degree of the sum and difference of the polynomials (3x^{5}y - 2x^{3}y^{4}-7xy^{3})…

Explanation:

Step1: Find the sum of the polynomials

Let (P(x,y)=3x^{5}y - 2x^{3}y^{4}-7xy^{3}) and (Q(x,y)=- 8x^{5}y+2x^{3}y^{4}+xy^{3}) (P(x,y)+Q(x,y)=(3x^{5}y - 2x^{3}y^{4}-7xy^{3})+(-8x^{5}y + 2x^{3}y^{4}+xy^{3})) [ \begin{align*} &=(3x^{5}y-8x^{5}y)+(-2x^{3}y^{4}+2x^{3}y^{4})+(-7xy^{3}+xy^{3})\ &=-5x^{5}y-6xy^{3} \end{align*} ] The degree of a term (ax^{m}y^{n}) is (m + n). For (-5x^{5}y), the degree is (5 + 1=6), for (-6xy^{3}), the degree is (1+3 = 4). So the degree of (P + Q) is (6)

Step2: Find the difference of the polynomials

(P(x,y)-Q(x,y)=(3x^{5}y - 2x^{3}y^{4}-7xy^{3})-(-8x^{5}y + 2x^{3}y^{4}+xy^{3})) [ \begin{align*} &=3x^{5}y - 2x^{3}y^{4}-7xy^{3}+8x^{5}y-2x^{3}y^{4}-xy^{3}\ &=(3x^{5}y + 8x^{5}y)+(-2x^{3}y^{4}-2x^{3}y^{4})+(-7xy^{3}-xy^{3})\ &=11x^{5}y-4x^{3}y^{4}-8xy^{3} \end{align*} ] For (11x^{5}y), the degree is (5+1 = 6), for (-4x^{3}y^{4}), the degree is (3 + 4=7), for (-8xy^{3}), the degree is (1+3=4). So the degree of (P - Q) is (7)

Answer:

The sum has a degree of (6), but the difference has a degree of (7) (the third option).