Find the degree of each of the following polynomials :
`(i) 3x^(4)-x^(2)+8 " " (ii) y^(2)-5y+7 " " (iii) 3x+4 " " (iv) 3`
`(v) x-2x^(2)+5x^(7) " " (vi) 2y^(2)-5y^(6)+1 " "(vii) x^(3)-1 " " (viii) 3x+5x^(5)+1`

Answer

To find the degree of each polynomial, we need to identify the highest power of the variable in each polynomial expression. The degree of a polynomial is defined as the highest exponent of the variable in the polynomial.

Let's solve the question step by step:

### Step 1: Identify the polynomials
We have the following polynomials to analyze:
1. \(3x^{4} - x^{2} + 8\)
2. \(y^{2} - 5y + 7\)
3. \(3x + 4\)
4. \(3\)
5. \(x - 2x^{2} + 5x^{7}\)
6. \(2y^{2} - 5y^{6} + 1\)
7. \(x^{3} - 1\)
8. \(3x + 5x^{5} + 1\)

### Step 2: Find the degree of each polynomial

**(i)** For the polynomial \(3x^{4} - x^{2} + 8\):
- The highest power of \(x\) is \(4\).
- **Degree = 4**

**(ii)** For the polynomial \(y^{2} - 5y + 7\):
- The highest power of \(y\) is \(2\).
- **Degree = 2**

**(iii)** For the polynomial \(3x + 4\):
- The highest power of \(x\) is \(1\).
- **Degree = 1**

**(iv)** For the polynomial \(3\):
- This is a constant polynomial, and the degree is \(0\).
- **Degree = 0**

**(v)** For the polynomial \(x - 2x^{2} + 5x^{7}\):
- The highest power of \(x\) is \(7\).
- **Degree = 7**

**(vi)** For the polynomial \(2y^{2} - 5y^{6} + 1\):
- The highest power of \(y\) is \(6\).
- **Degree = 6**

**(vii)** For the polynomial \(x^{3} - 1\):
- The highest power of \(x\) is \(3\).
- **Degree = 3**

**(viii)** For the polynomial \(3x + 5x^{5} + 1\):
- The highest power of \(x\) is \(5\).
- **Degree = 5**

### Summary of Degrees
1. \(3x^{4} - x^{2} + 8\) → Degree = 4
2. \(y^{2} - 5y + 7\) → Degree = 2
3. \(3x + 4\) → Degree = 1
4. \(3\) → Degree = 0
5. \(x - 2x^{2} + 5x^{7}\) → Degree = 7
6. \(2y^{2} - 5y^{6} + 1\) → Degree = 6
7. \(x^{3} - 1\) → Degree = 3
8. \(3x + 5x^{5} + 1\) → Degree = 5