$\alpha$ and $\beta$ are zeroes of a quadratic polynomial $px^2 + qx + 1$. Form a quadratic polynomial whose zeroes are $\dfrac{2}{\alpha}$ and $\dfrac{2}{\beta}$.
Generated by claude-sonnet-4-6 · 2026-06-15 10:36 · grounding rag
Model Answer
For $px^2 + qx + 1$, by Vieta's formulas:
$$\alpha + \beta = \frac{-q}{p}, \qquad \alpha\beta = \frac{1}{p}$$
New zeroes are $\dfrac{2}{\alpha}$ and $\dfrac{2}{\beta}$.
Sum of new zeroes:
$$\frac{2}{\alpha} + \frac{2}{\beta} = \frac{2(\alpha+\beta)}{\alpha\beta} = \frac{2 \cdot \left(\dfrac{-q}{p}\right)}{\dfrac{1}{p}} = -2q$$
Product of new zeroes:
$$\frac{2}{\alpha} \times \frac{2}{\beta} = \frac{4}{\alpha\beta} = \frac{4}{\dfrac{1}{p}} = 4p$$
Required quadratic polynomial:
$$k\left[x^2 - (\text{sum})x + \text{product}\right] = k\left[x^2 + 2qx + 4p\right]$$
Taking $k = 1$: $\boxed{x^2 + 2qx + 4p}$
Source: Chapter 2, Section 2.3
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Explanation
- First find $\alpha+\beta$ and $\alpha\beta$ from the given polynomial using $\frac{-b}{a}$ and $\frac{c}{a}$ (here $a=p,\ b=q,\ c=1$).
- Then compute sum and product of the new zeroes $\frac{2}{\alpha}$ and $\frac{2}{\beta}$ by expressing them in terms of $\alpha+\beta$ and $\alpha\beta$.
- Finally use the standard form $x^2 - (\text{sum})x + \text{product}$ to write the polynomial.
- Examiners award 1 mark each for: correct sum, correct product, and correct final polynomial.