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: Find the difference of polynomials

Subtract the second polynomial from the first:
$$(a^3b + 9a^2b^2 - 4ab^5) - (a^3b - 3a^2b^2 + ab^5)$$
Distribute the negative sign:
$$a^3b + 9a^2b^2 - 4ab^5 - a^3b + 3a^2b^2 - ab^5$$

Step2: Combine like terms

• For \(a^3b\): \(a^3b - a^3b = 0\)
• For \(a^2b^2\): \(9a^2b^2 + 3a^2b^2 = 12a^2b^2\)
• For \(ab^5\): \(-4ab^5 - ab^5 = -5ab^5\)

The simplified difference is \(12a^2b^2 - 5ab^5\), which is a binomial (two terms).

Step3: Determine the degree

The degree of a term is the sum of exponents.

• For \(12a^2b^2\): \(2 + 2 = 4\)
• For \(-5ab^5\): \(1 + 5 = 6\)

The highest degree is 6.

Answer:

The difference is a binomial with a degree of 6. (Option: The difference is a binomial with a degree of 6.)