If `alpha and beta` are the zeros of the quadratic polynomial `f(x)=a x^2+b x+c ,` then evaluate: (i) `alpha^2+beta^2 `alpha/beta+beta/alpha` alpha^3+beta^3` `1/(alpha^3)+1/(beta^3)` (alpha^2)/beta+(beta^2)/alpha`

Explanation

To solve the problem step by step, we will use the relationships between the roots of the quadratic polynomial \( f(x) = ax^2 + bx + c \), where \( \alpha \) and \( \beta \) are the roots.

### Step 1: Calculate \( \alpha^2 + \beta^2 \)

Using the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
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