Question

If the zeroes of the polynomial $f\left(x\right)=x^3-3x^2+x+1$ are $a-b$ , $a$, $a+b$ , find a and b .

Answer 1

Zeroes of $x^3-3x^2+x+1$ are $a-b,a,a+b$

$\therefore$ sum of zeroes $=a-b+a+a+b=3\left(\frac{coefficientof{(-x)}^2}{coefficientof{x}^3}\right)$

$\Rightarrow 3a=3$

$\Rightarrow a=1$

Product of zeroes $=-1$

$\Rightarrow (a-b)(a+b)a=-1$

$\Rightarrow (1-b)(1+b)=-1$

$\Rightarrow 1-b^2=-1$

$\Rightarrow b^2=2$

$\Rightarrow b=\pm \sqrt{2}.$

Hence, the answer is $\pm \sqrt{2}.$