Question

If the zeros of the polynomials $f\left(x\right)=2x^3-15x^2+37x-30$ are in A. P., find them.

Answer 1

Let the roots be $a-d,a,a+d$ (as in A.P)

$f\left(x\right)=2x^3-15x^2+37x-30$

Sum=$a-d+a+a+d=\frac{15}{2}$

$3a=\frac{15}{2}$

$a=\frac{5}{2}$

Product$=(a-d)\left(a\right)(a+d)=\frac{30}{2}$

$(a^2-d^2)a=15$

$(a^2-d^2)\frac{5}{2}=15$

$a^2-d^2=6$

$d^2={\left(\frac{5}{2}\right)}^2-6$

$d^2=\frac{25}{4}-6$

$d^2=\frac{1}{4}$

$\Rightarrow d{=}_{-}^{+}\frac{1}{2}.$

$\therefore$ Roots are $2,3,\frac{5}{2}$