Let the quadratic polynomial be represented as:

$p(x) = ax^2 + bx + c$

Given that $\alpha$ and $\beta$ are the zeroes of this polynomial, we know the following relationships from Vieta's formulas:

1. The sum of the roots $\alpha + \beta = -\frac{b}{a}$
2. The product of the roots $\alpha \beta = \frac{c}{a}$

We are asked to find a new quadratic polynomial whose zeroes are $\frac{2\alpha}{\beta}$ and $\frac{2\beta}{\alpha}$.

To do this, we need to compute:

• The sum of the new zeroes:
  $\text{Sum of new roots} = \frac{2\alpha}{\beta} + \frac{2\beta}{\alpha} = 2\left( \frac{\alpha}{\beta} + \frac{\beta}{\alpha} \right)$ Using the identity $\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^2 + \beta^2}{\alpha \beta}$, we can express the sum as: $\text{Sum of new roots} = 2\left( \frac{\alpha^2 + \beta^2}{\alpha \beta} \right)$ Now, recall that: $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$ Substitute $\alpha + \beta = -\frac{b}{a}$ and $\alpha \beta = \frac{c}{a}$ into the equation: $\alpha^2 + \beta^2 = \left( -\frac{b}{a} \right)^2 - 2\left( \frac{c}{a} \right)$ $\alpha^2 + \beta^2 = \frac{b^2}{a^2} - \frac{2c}{a}$ Thus, the sum of the new roots becomes: $\text{Sum of new roots} = 2 \cdot \frac{\frac{b^2}{a^2} - \frac{2c}{a}}{\frac{c}{a}} = 2 \cdot \frac{b^2 - 2ac}{ac}$

• The product of the new zeroes:
  $\text{Product of new roots} = \frac{2\alpha}{\beta} \cdot \frac{2\beta}{\alpha} = \frac{4\alpha\beta}{\alpha\beta} = 4$

Thus, the new quadratic polynomial has the form:

$p(x) = x^2 - (\text{Sum of new roots}) \cdot x + (\text{Product of new roots})$

Substituting the values for the sum and product:

$p(x) = x^2 - \left( 2 \cdot \frac{b^2 - 2ac}{ac} \right) x + 4$

This is the required quadratic polynomial.

---

Would you like further details or have any questions on the steps?

Here are five related questions you might find helpful:

1. What are Vieta's formulas and how do they relate to the zeroes of a polynomial?
2. How do you simplify algebraic expressions involving fractions like $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$?
3. Can you derive the relationship between the sum of the squares of the roots and the sum and product of the roots in a quadratic polynomial?
4. How can you verify that the quadratic polynomial you derived has the correct zeroes?
5. What would happen to the polynomial if the coefficients $a$, $b$, and $c$ were given specific values?

Tip: To simplify problems involving fractions with variables, always look for common denominators or factorization strategies!