Question

which algebraic expression is a polynomial with a degree of 5?
○ $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$
○ $2xy^4 + 4x^2y^3 - 6x^3y^2 - 7x^4$
○ $8y^6 + y^5 - 5xy^3 + 7x^2y^2 - x^3y - 6x^4$
○ $-6xy^5 + 5x^2y^3 - x^3y^2 + 2x^2y^3 - 3xy^5$

Explanation:

Step1: Recall polynomial degree rule

The degree of a term in a multivariable polynomial is the sum of the exponents of its variables. The degree of the polynomial is the highest degree of its individual terms.

Step2: Calculate degree of Option1

For $3x^5 + 8x^4y^2 - 9x^3y^3 - 6y^5$:

• Degree of $3x^5$: $5$
• Degree of $8x^4y^2$: $4+2=6$
• Degree of $-9x^3y^3$: $3+3=6$
• Degree of $-6y^5$: $5$

Highest degree = $6$

Step3: Calculate degree of Option2

For $2xy^4 + 4x^2y^3 - 6x^3y^2 - 7x^4$:

• Degree of $2xy^4$: $1+4=5$
• Degree of $4x^2y^3$: $2+3=5$
• Degree of $-6x^3y^2$: $3+2=5$
• Degree of $-7x^4$: $4$

Highest degree = $5$

Step4: Calculate degree of Option3

For $8y^6 + y^5 - 5xy^3 + 7x^2y^2 - x^3y - 6x^4$:

• Degree of $8y^6$: $6$
• Degree of $y^5$: $5$
• Degree of $-5xy^3$: $1+3=4$
• Degree of $7x^2y^2$: $2+2=4$
• Degree of $-x^3y$: $3+1=4$
• Degree of $-6x^4$: $4$

Highest degree = $6$

Step5: Calculate degree of Option4

For $-6xy^5 + 5x^2y^3 - x^3y^2 + 2x^2y^3 - 3xy^5$:

• Degree of $-6xy^5$: $1+5=6$
• Degree of $5x^2y^3$: $2+3=5$
• Degree of $-x^3y^2$: $3+2=5$
• Degree of $2x^2y^3$: $2+3=5$
• Degree of $-3xy^5$: $1+5=6$

Highest degree = $6$

Answer:

B. $2xy^4 + 4x^2y^3 - 6x^3y^2 - 7x^4$