The degree of the polynomial `3a ^(2) + 4b ^(3) - 5 ab ^(3) + 7 - 5a ^(7) b ^(5)` is

Options

A. 2
B. 3
C. 4
D. 12

Answer

To find the degree of the polynomial \(3a^2 + 4b^3 - 5ab^3 + 7 - 5a^7b^5\), we will analyze each term in the polynomial and calculate the sum of the exponents for each term.

### Step-by-Step Solution:

1. **Identify each term in the polynomial:**
The polynomial consists of the following terms:
- \(3a^2\)
- \(4b^3\)
- \(-5ab^3\)
- \(7\)
- \(-5a^7b^5\)

2. **Calculate the degree of each term:**
- For the term \(3a^2\):
- The exponent of \(a\) is \(2\) and there is no \(b\), so the degree is \(2\).
- For the term \(4b^3\):
- The exponent of \(b\) is \(3\) and there is no \(a\), so the degree is \(3\).
- For the term \(-5ab^3\):
- The exponent of \(a\) is \(1\) and the exponent of \(b\) is \(3\). The total degree is \(1 + 3 = 4\).
- For the term \(7\):
- This is a constant term, so the degree is \(0\).
- For the term \(-5a^7b^5\):
- The exponent of \(a\) is \(7\) and the exponent of \(b\) is \(5\). The total degree is \(7 + 5 = 12\).

3. **Determine the highest degree:**
Now, we compare the degrees of all the terms:
- Degree of \(3a^2\) is \(2\)
- Degree of \(4b^3\) is \(3\)
- Degree of \(-5ab^3\) is \(4\)
- Degree of \(7\) is \(0\)
- Degree of \(-5a^7b^5\) is \(12\)

The highest degree among these is \(12\).

### Conclusion:
The degree of the polynomial \(3a^2 + 4b^3 - 5ab^3 + 7 - 5a^7b^5\) is **12**.

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