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 \( b \) in the polynomial \( ax^2 - 7x + b \) given that the zeroes are \( x = \frac{5}{3} \) 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: Use the relationship between roots and coefficients
For a quadratic polynomial \( x^2 - (\alpha + \beta)x + \alpha \beta = 0 \), we can express the polynomial in terms of its roots:
- The sum of the roots \( \alpha + \beta \) is:
\[ \alpha + \beta = \frac{5}{3} + \left(-\frac{1}{2}\right) = \frac{5}{3} - \frac{1}{2} \]
To add these fractions, we need a common denominator, which is 6:
\[ \frac{5}{3} = \frac{10}{6}, \quad -\frac{1}{2} = -\frac{3}{6} \]
Therefore,
\[ \alpha + \beta = \frac{10}{6} - \frac{3}{6} = \frac{7}{6} \]

- The product of the roots \( \alpha \beta \) is:
\[ \alpha \beta = \frac{5}{3} \cdot \left(-\frac{1}{2}\right) = -\frac{5}{6} \]

### Step 3: Form the polynomial using the roots
Using the roots, we can write the polynomial as:
\[ x^2 - \left(\frac{7}{6}\right)x - \left(-\frac{5}{6}\right) = 0 \]
This simplifies to:
\[ x^2 - \frac{7}{6}x - \frac{5}{6} = 0 \]

### Step 4: Eliminate the fractions
To eliminate the fractions, we multiply the entire equation by 6:
\[ 6x^2 - 7x - 5 = 0 \]

### Step 5: Compare coefficients
Now we compare this polynomial with the given polynomial \( ax^2 - 7x + b \):
- From \( 6x^2 \), we see that \( a = 6 \).
- From \( -5 \), we see that \( b = -5 \).

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