If the zero of the polynomial $x^3-2 x^2+x+1$ are a-b, a and a+b, then find the values of a and b.
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If the zero of the polynomial $x^3-2 x^2+x+1$ are $a-b, a$ and $a+b$, then find the values of $a$ and $b$.

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SOLUTION —

As $(a-b), a$ and $(a+b)$ are zeros of $x^3-3 x^2+x+1$,

we have $: a-b+a+a+b=3$

$\Rightarrow 3 a=3$ or, $a=1$ ..... (i)

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

$\Rightarrow 3 a^2-b^2=1$ ........(ii)

and $(a-b) a(a+b)=-1$

$\Rightarrow a\left(a^2-b^2\right)=-1$ ........ (iii)

From (i) and (ii), we have

$b= \pm \sqrt{2}$

Thus, $a=1, b= \pm \sqrt{2}$

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