Which is true about the completely simplified difference of the polynomials *a*3*b* + 9*a*2*b*2 − 4*ab*5 and *a*3*b* − 3*a*2*b*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.

To determine the correct statement about the difference of the two polynomials, let's first calculate the difference and then simplify it.

### Given polynomials:

1. \( a^3b + 9a^2b^2 - 4ab^5 \)
2. \( a^3b - 3a^2b^2 + ab^5 \)

### Step 1: 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:

\[
a^3b + 9a^2b^2 - 4ab^5 - a^3b + 3a^2b^2 - ab^5
\]

### Step 2: Combine like terms

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

So, the completely simplified difference is:

\[
12a^2b^2 - 5ab^5
\]

### Step 3: Determine the number of terms and the degree

- The resulting polynomial is \( 12a^2b^2 - 5ab^5 \).
- It has **two terms**, so it is a **binomial**.
- The degree of each term:
  - \( 12a^2b^2 \) has a degree of \( 2 + 2 = 4 \).
  - \( -5ab^5 \) has a degree of \( 1 + 5 = 6 \).

### Conclusion:
- The difference is a **binomial** with a **degree of 6**.

Therefore, the correct statement is: **The difference is a binomial with a degree of 6.**

Would you like more details on this or have any questions? Here are some related questions:

1. How do you determine the degree of a polynomial when it contains multiple variables?
2. What are the steps to subtract one polynomial from another?
3. How can identifying like terms help in simplifying polynomials?
4. What is the significance of the highest degree in a polynomial?
5. Can a binomial ever have a higher degree than a trinomial?

**Tip:** Always pay attention to the signs when subtracting polynomials to ensure accurate simplification.

Explore the difference of two polynomials, including step-by-step simplification and determination of the polynomial's degree. Learn the concepts of polynomials, subtraction of polynomials, and degree determination. Suitable for high school students.

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