What cannot be the exact number of real roots in a polynomial of degree 9?

Options

A. 5
B. 6
C. 7
D. 9

Answer

To determine what cannot be the exact number of real roots in a polynomial of degree 9, we need to analyze the properties of polynomial roots.

### Step-by-Step Solution:

1. **Understanding the Degree of the Polynomial**:
A polynomial of degree 9 can have a total of 9 roots. These roots can be either real or complex (imaginary).

**Hint**: Recall that the degree of a polynomial indicates the maximum number of roots it can have.

2. **Types of Roots**:
- Real roots: These are the roots that can be plotted on the number line.
- Complex roots: These occur in conjugate pairs. For example, if \( a + bi \) is a root, then \( a - bi \) is also a root.

**Hint**: Remember that complex roots come in pairs, which affects the count of real roots.

3. **Counting Real Roots**:
Since complex roots come in pairs, the number of complex roots must be even. Therefore, if we denote the number of complex roots as \( 2k \) (where \( k \) is a non-negative integer), the number of real roots \( r \) can be expressed as:
\[ r = 9 - 2k \]
Here, \( r \) must be a non-negative integer.

**Hint**: The total number of roots (real + complex) must equal the degree of the polynomial.

4. **Determining Possible Values for Real Roots**:
Since \( k \) can take values from 0 up to 4 (because \( 2k \) must be less than or equal to 9), we can find the possible values for \( r \):
- If \( k = 0 \): \( r = 9 - 0 = 9 \) (all roots are real)
- If \( k = 1 \): \( r = 9 - 2 = 7 \) (7 real roots)
- If \( k = 2 \): \( r = 9 - 4 = 5 \) (5 real roots)
- If \( k = 3 \): \( r = 9 - 6 = 3 \) (3 real roots)
- If \( k = 4 \): \( r = 9 - 8 = 1 \) (1 real root)

**Hint**: List the values of \( r \) as you calculate them based on the possible values of \( k \).

5. **Identifying the Impossible Count**:
The possible counts of real roots derived from the above calculations are: 9, 7, 5, 3, and 1. The only even number in the options given (5, 6, 7, 9) is 6. Since the number of real roots must be odd (as derived from the calculations), 6 cannot be the exact number of real roots in a polynomial of degree 9.

**Hint**: Check the parity (odd/even) of the possible counts to identify which number cannot occur.

### Conclusion:
The exact number of real roots that cannot occur in a polynomial of degree 9 is **6**.