Question

If α and β are the zeros of the polynomial $2x^2+x-6,$ then form a quadratic equation whose zeros are 2α and 2β.

Answer 1

f(x) = 2x² + x - 6 is the given polynomial.
It is given that,α and β are the zeros of the polynomial.
$\alpha +\beta =\frac{-1}{2}\mathrm{and}\mathrm{\alpha \beta }=\frac{-6}{2}=-3$
Let 2α and 2β be the zeros of the polynomial g(x).
Thus,
Sum of roots = 2α and 2β = 2(α + β) = $\left(2\times \frac{-1}{2}\right)=-1$
Product of roots = 2α × 2β = 4αβ = 4(- 3) = - 12

The required polynomial g(x) = x² - (Sum of roots)x + Product of roots
= x² - (- 1)x - 12 = x² + x - 12
= x² + x - 12