If `x=(5)/(3)` and `x=-(1)/(2)` are the zeroes of the polynomial `ax^(2)-7x+b`, then find the values of a and b.

Explanation

To find the values of \( a \) and \( b \) for the polynomial \( ax^2 - 7x + b \) given the roots \( x = \frac{5}{3} \) and \( x = -\frac{1}{2} \), we can follow these steps:

### Step 1: Identify the roots
The roots of the polynomial are given as:
- \( \alpha = \frac{5}{3} \)
- \( \beta = -\frac{1}{2} \)

### Step 2: Calculate \( \alpha + \beta \)
To find \( \alpha + \beta \):
\[ \alpha + \beta = \frac{5}{3} + \left(-\frac{1}{2}\right) \]
To add these fractions, we need a common denominator. The least common multiple of 3 and 2 is 6.
\[ \alpha + \beta = \frac{5 \times 2}{3 \times 2} + \left(-\frac{1 \times 3}{2 \times 3}\right) = \frac{10}{6} - \frac{3}{6} = \frac{7}{6} \]

### Step 3: Calculate \( \alpha \beta \)
Now, calculate \( \alpha \beta \):
\[ \alpha \beta = \frac{5}{3} \times \left(-\frac{1}{2}\right) = -\frac{5}{6} \]

### Step 4: Write the polynomial in terms of its roots
The polynomial with roots \( \alpha \) and \( \beta \) can be expressed as:
\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]
Substituting the values we found:
\[ x^2 - \left(\frac{7}{6}\right)x - \frac{5}{6} = 0 \]

### Step 5: Multiply through by 6 to eliminate fractions
To eliminate the fractions, multiply the entire equation by 6:
\[ 6x^2 - 7x - 5 = 0 \]

### Step 6: Compare with the given polynomial
The given polynomial is:
\[ ax^2 - 7x + b = 0 \]
Comparing coefficients, we find:
- Coefficient of \( x^2 \): \( a = 6 \)
- Constant term: \( b = -5 \)

### Final Answer
Thus, the values of \( a \) and \( b \) are:
\[ a = 6, \quad b = -5 \]