Question

which is true about the completely simplified difference of the polynomials ( a^3b + 9a^2b^2 - 4ab^5 ) and ( a^3b - 3a^2b^2 + ab^5 )?

• the difference is a binomial with a degree of 5.
• the difference is a binomial with a degree of 6.
• the difference is a trinomial with a degree of 5.
• the difference is a trinomial with a degree of 6.

Explanation:

Step1: Define the polynomial difference

Let $P_1 = a^3b + 3a^2b^2 - 4ab^5$ and $P_2 = a^3b - 3a^2b^2 + ab^5$.
The difference is $P_1 - P_2 = (a^3b + 3a^2b^2 - 4ab^5) - (a^3b - 3a^2b^2 + ab^5)$

Step2: Distribute the negative sign

$P_1 - P_2 = a^3b + 3a^2b^2 - 4ab^5 - a^3b + 3a^2b^2 - ab^5$

Step3: Combine like terms

• For $a^3b$ terms: $a^3b - a^3b = 0$
• For $a^2b^2$ terms: $3a^2b^2 + 3a^2b^2 = 6a^2b^2$
• For $ab^5$ terms: $-4ab^5 - ab^5 = -5ab^5$

Result: $P_1 - P_2 = 6a^2b^2 - 5ab^5$

Step4: Classify the polynomial

Count terms: 2 terms (binomial).
Find degree: For $6a^2b^2$, degree $2+2=4$; for $-5ab^5$, degree $1+5=6$. The highest degree is 6.

Answer:

The difference is a binomial with a degree of 6.