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

which is true about the degree of the sum and difference of the polynomials $3x^{5}y - 2x^{3}y^{4}-7xy^{3}$ and $-8x^{5}y + 2x^{3}y^{4}+xy^{3}$?\nboth the sum and difference have a degree of 6.\nboth the sum and difference have a degree of 7.\nthe sum has a degree of 6, but the difference has a degree of 7.\nthe sum has a degree of 7, but the difference has a degree of 6.

which is true about the degree of the sum and difference of the polynomials $3x^{5}y - 2x^{3}y^{4}-7xy^{3}$ and $-8x^{5}y + 2x^{3}y^{4}+xy^{3}$?\nboth the sum and difference have a degree of 6.\nboth the sum and difference have a degree of 7.\nthe sum has a degree of 6, but the difference has a degree of 7.\nthe sum has a degree of 7, but the difference has a degree of 6.

Answer

Explanation:

Step1: Find the sum of the polynomials

Let (P = 3x^{5}y-2x^{3}y^{4}-7xy^{3}) and (Q=-8x^{5}y + 2x^{3}y^{4}+xy^{3}). Then (P + Q=(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})=-5x^{5}y - 6xy^{3}). 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-Q=(3x^{5}y-2x^{3}y^{4}-7xy^{3})-(-8x^{5}y + 2x^{3}y^{4}+xy^{3})=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}). The degree of (11x^{5}y) is (5 + 1=6), the degree of (-4x^{3}y^{4}) is (3+4 = 7), the degree of (-8xy^{3}) 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.