Question

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

Explanation:

Step1: Define the two polynomials

Let $P(x,y) = 3x^5y - 2x^3y^4 - 7xy^3$
Let $Q(x,y) = -8x^5y + 2x^3y^4 + xy^3$

Step2: Calculate the sum $P+Q$

Combine like terms:

$$\begin{align*} P(x,y)+Q(x,y)&=(3x^5y-8x^5y)+(-2x^3y^4+2x^3y^4)+(-7xy^3+xy^3)\\ &=-5x^5y + 0x^3y^4 -6xy^3\\ &=-5x^5y -6xy^3 \end{align*}$$

Find the degree of each term:

• Degree of $-5x^5y$: $5+1=6$
• Degree of $-6xy^3$: $1+3=4$

The highest degree is 6.

Step3: Calculate the difference $P-Q$

Distribute the negative sign and combine like terms:

$$\begin{align*} P(x,y)-Q(x,y)&=(3x^5y-(-8x^5y))+(-2x^3y^4-2x^3y^4)+(-7xy^3-xy^3)\\ &=(3x^5y+8x^5y)+(-2x^3y^4-2x^3y^4)+(-7xy^3-xy^3)\\ &=11x^5y -4x^3y^4 -8xy^3 \end{align*}$$

Find the degree of each term:

• Degree of $11x^5y$: $5+1=6$
• Degree of $-4x^3y^4$: $3+4=7$
• Degree of $-8xy^3$: $1+3=4$

The highest degree is 7.

Answer:

The sum has a degree of 6, but the difference has a degree of 7.