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Question

If the zeros of the polynomial f(x)=x33x2+x+1 are  (a-b), a and (a+b), find a and b.


Solution

As the the coefficient of highest power x3 is1, if three roots are α,βandγ, we have

f(x)=x33x2+x+1

=x3(α+β+γ)x2+(αβ+γβ+γα)x+αβγ

Now let us compare coefficients of similar powers on each side. 

First comparing the sum of roots from the coefficient of x2, we have

α+β+γ=ba

ab+a+a+b=(31)

i.e. 3a=3

i.e.a=1.

Also, coefficients of x give us the sum of the product of zeros

αβ+γβ+γα=ca=1

a(ab)+a(a+b)+(a+b)(ab)=1

i.e. a(ab+a+b)+a2b2=1

=2a2+a2b2
=3a2b2=1

and as 
a=1,
b2=3×(1)21=2 and 

b=±2

Further, products of roots is (ab)a(a+b)=1.

Hence, a=1 and b=±2

Note that the two values of b give the same set of roots.

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