1. **State the problem:** We are given two zeroes $\alpha$ and $\beta$ of a quadratic polynomial such that $\alpha + \beta = 24$ and $\alpha - \beta = 8$. We need to find the quadratic polynomial. 2. **Use the given information:** We have two equations: $$\alpha + \beta = 24$$ $$\alpha - \beta = 8$$ 3. **Find $\alpha$ and $\beta$ by solving the system:** Add the two equations: $$ (\alpha + \beta) + (\alpha - \beta) = 24 + 8 $$ $$ 2\alpha = 32 $$ $$ \alpha = \frac{\cancel{2}\times 16}{\cancel{2}} = 16 $$ Substitute $\alpha = 16$ into $\alpha + \beta = 24$: $$ 16 + \beta = 24 $$ $$ \beta = 24 - 16 = 8 $$ 4. **Form the quadratic polynomial:** If $\alpha$ and $\beta$ are roots, the polynomial is: $$ x^2 - (\alpha + \beta)x + \alpha\beta $$ Calculate $\alpha\beta$: $$ 16 \times 8 = 128 $$ So the polynomial is: $$ x^2 - 24x + 128 $$ 5. **Final answer:** The quadratic polynomial with roots $\alpha$ and $\beta$ is: $$ x^2 - 24x + 128 $$