If `a` and `b` are randomly chosen from the set `{1,2,3,4,5,6,7,8,9}`, then the probability that the expression `ax^(4)+bx^(3)+(a+1)x^(2)+bx+1` has positive values for all real values of `x` is

Options

A. `(34)/(81)`
B. `(31)/(81)`
C. `(32)/(81)`
D. `(10)/(27)`

Explanation

To find the probability that the expression \( ax^4 + bx^3 + (a+1)x^2 + bx + 1 \) has positive values for all real values of \( x \), we need to analyze the conditions under which this polynomial is always positive.

### Step-by-Step Solution:

1. **Identify the Polynomial**:
The given polynomial is \( P(x) = ax^4 + bx^3 + (a+1)x^2 + bx + 1 \).

2. **Condition for Positivity**:
A polynomial is positive for all real \( x \) if it has no real roots. This occurs when the discriminant of the quadratic form of the polynomial is less than zero.

3. **Rewrite the Polynomial**:
We can express the polynomial in a quadratic form in terms of \( x^2 \):
\[ P(x) = a(x^4) + b(x^3) + (a+1)(x^2) + b(x) + 1 \]
We can treat \( x^2 \) as a new variable \( y \) (where \( y = x^2 \)), leading to the quadratic form:
\[ P(y) = ay^2 + by + (a + 1) \]

4. **Calculate the Discriminant**:
The discriminant \( D \) of a quadratic \( Ay^2 + By + C \) is given by:
\[ D = B^2 - 4AC \]
For our polynomial:
- \( A = a \)
- \( B = b \)
- \( C = a + 1 \)

Therefore, the discriminant is:
\[ D = b^2 - 4a(a + 1) \]

5. **Set the Condition for Non-Real Roots**:
For the polynomial to be positive for all \( x \), we need:
\[ D < 0 \implies b^2 < 4a(a + 1) \]

6. **Count Valid Pairs of \( (a, b) \)**:
We need to count the pairs \( (a, b) \) such that \( a \) and \( b \) are chosen from the set \( \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \) and satisfy the condition \( b^2 < 4a(a + 1) \).

- For each value of \( a \), determine the maximum \( b \) that satisfies the inequality.

| \( a \) | Maximum \( b \) satisfying \( b^2 < 4a(a + 1) \) | Valid \( b \) values |
|---------|----------------------------------------------------|----------------------|
| 1 | 1 | 1 |
| 2 | 2 | 1, 2 |
| 3 | 3 | 1, 2, 3 |
| 4 | 3 | 1, 2, 3 |
| 5 | 4 | 1, 2, 3, 4 |
| 6 | 4 | 1, 2, 3, 4 |
| 7 | 5 | 1, 2, 3, 4, 5 |
| 8 | 5 | 1, 2, 3, 4, 5 |
| 9 | 5 | 1, 2, 3, 4, 5 |

Now, count the valid pairs:
- For \( a = 1 \): 1 valid \( b \) → 1 pair
- For \( a = 2 \): 2 valid \( b \) → 2 pairs
- For \( a = 3 \): 3 valid \( b \) → 3 pairs
- For \( a = 4 \): 3 valid \( b \) → 3 pairs
- For \( a = 5 \): 4 valid \( b \) → 4 pairs
- For \( a = 6 \): 4 valid \( b \) → 4 pairs
- For \( a = 7 \): 5 valid \( b \) → 5 pairs
- For \( a = 8 \): 5 valid \( b \) → 5 pairs
- For \( a = 9 \): 5 valid \( b \) → 5 pairs

Total valid pairs = \( 1 + 2 + 3 + 3 + 4 + 4 + 5 + 5 + 5 = 32 \).

7. **Calculate Total Outcomes**:
The total number of outcomes when choosing \( a \) and \( b \) from the set is \( 9 \times 9 = 81 \).

8. **Calculate the Probability**:
The probability that the polynomial is positive for all real \( x \) is given by:
\[ P = \frac{\text{Number of valid pairs}}{\text{Total pairs}} = \frac{32}{81} \]

### Final Answer:
The probability that the expression \( ax^4 + bx^3 + (a+1)x^2 + bx + 1 \) has positive values for all real values of \( x \) is \( \frac{32}{81} \).