Question

Find the polynomials whose zeroes are three times the zeroes of $2x^2-3x+1$.

Answer 1

zeros of $2x^2-3x+1$

$\Rightarrow 2x^2-3x+1=0$

$\Rightarrow 2x^2-2x-x+1=0$

$\Rightarrow 2x(x-1)-(x-1)=0$

$\Rightarrow (x-1)(2x-1)=0$

$\therefore$ zeros of $2x^2-3x+1$ are $(1,1/2)$

three times zeros of $2x^2-3x+1\Rightarrow (3,3/2)$

polynomial with $(3,3/2)$ as zeros

$\Rightarrow (x-3)(x-3/2)$

$\Rightarrow x^2-\frac{3x}{2}-3x+\frac{9}{2}$

$\Rightarrow x^2-\frac{-9x}{2}+\frac{9}{2}$

$\Rightarrow 2x^2-9x+9$

$\therefore$ the set of polynomials whose zeros are 3 times zeros of $2x^2-3x+1$ are

$k(2x^2-9x+9)$ where k = constant