If `alpha,beta` are roots of `x^(2)-px+q=0`, find the value of
(i) `alpha^(2)+beta^(2)`
(ii) `alpha^(3)+beta^(3)`
(iii) `alpha-beta`,
(iv) `alpha^(4)+beta^(4)`.

Answer

To solve the given problem, we will use the relationships between the roots of the quadratic equation and the coefficients. The roots \( \alpha \) and \( \beta \) of the equation \( x^2 - px + q = 0 \) satisfy the following relationships:

1. \( \alpha + \beta = p \)
2. \( \alpha \beta = q \)

Now, we will find the required values step by step.

### (i) Finding \( \alpha^2 + \beta^2 \)

We can use the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]

Substituting the known values:
\[ \alpha^2 + \beta^2 = p^2 - 2q \]

### (ii) Finding \( \alpha^3 + \beta^3 \)

We can use the identity:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 + \beta^2 - \alpha\beta) \]

Substituting the known values:
\[ \alpha^3 + \beta^3 = p\left((p^2 - 2q) - q\right) = p(p^2 - 3q) \]

### (iii) Finding \( \alpha - \beta \)

We can use the identity:
\[ \alpha - \beta = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} \]

Substituting the known values:
\[ \alpha - \beta = \sqrt{p^2 - 4q} \]

### (iv) Finding \( \alpha^4 + \beta^4 \)

We can use the identity:
\[ \alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2(\alpha\beta)^2 \]

Substituting the known values:
\[ \alpha^4 + \beta^4 = (p^2 - 2q)^2 - 2q^2 \]

Expanding this:
\[ = p^4 - 4pq + 4q^2 - 2q^2 = p^4 - 4pq + 2q^2 \]

### Summary of Results

1. \( \alpha^2 + \beta^2 = p^2 - 2q \)
2. \( \alpha^3 + \beta^3 = p(p^2 - 3q) \)
3. \( \alpha - \beta = \sqrt{p^2 - 4q} \)
4. \( \alpha^4 + \beta^4 = p^4 - 4pq + 2q^2 \)