Given: The polynomial $ax^2 + 7x + b$

To find: The values of $a$ and $b$ when the zeroes of the polynomial are $\frac{2}{3}$ and $-3$.

Solution:

To find the values of $a$ and $b$, we can use the fact that the sum and product of the zeroes of a quadratic polynomial can be expressed in terms of its coefficients.

Sum of the zeroes:
The sum of the zeroes of the quadratic polynomial $ax^2 + 7x + b$ is given by the formula:
$-\frac{b}{a} = \frac{2}{3} - 3$

Simplifying the equation, we get:
$-\frac{b}{a} = \frac{2}{3} - \frac{9}{3}$
$-\frac{b}{a} = -\frac{7}{3}$

Product of the zeroes:
The product of the zeroes of the quadratic polynomial $ax^2 + 7x + b$ is given by the formula:
\(\frac{c}{a} = \frac{2}{3} \times -3\)

Simplifying the equation, we get:
\(\frac{c}{a} = -2\)

Finding the value of a:
From the equation \(-\frac{b}{a} = -\frac{7}{3}\), we can equate the denominators and solve for a:
\(3b = 7a\)

We can assume a value for b, say b = 7, and solve for a:
\(3 \times 7 = 7a\)
\(21 = 7a\)
\(a = 3\)

Finding the value of b:
Substituting the value of a in the equation \(3b = 7a\), we can solve for b:
\(3b = 7 \times 3\)
\(3b = 21\)
\(b = 7\)

Conclusion:
Therefore, the values of a and b in the polynomial $ax^2 + 7x + b$ when the zeroes are $\frac{2}{3}$ and $-3$ are a = 3 and b = 7.