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If a-b,a, a+b are zero of polynomial x^3 -3x^2 +x+1, then find

Question

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

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Theodore Discussed

a - b, a, a + b

are zero of polynomial

x^{3} - 3 x^{2} + x + 1

, then find

a

and

b

10 mins ago

Discuss this question LIVE

10 mins ago

Text solutionVerified

Given polynomial

= x^{3} - 3 x^{2} + x + 1

Comparing it with

A x^{3} + B x^{2} + C x + D

A = 1, B = - 3, C = 1

and

D = 1

Sum of zeroes

= - \frac{B}{A} = - (\frac{- 3}{1}) = 3

But zeros are

a - b, a

and

a + b

Thus

a - b + a + a + b = 3

\Rightarrow 3 a = 3 \Rightarrow a = \frac{3}{3} = 1

Product of zeroes

= - \frac{D}{A} = - (\frac{1}{1}) = - 1

But product of zeros

= (a - b) (a) (a + b) = a (a^{2} - b^{2})

Then

a (a^{2} - b^{2}) = - 1

∴ 1 (1^{2} - b^{2}) = - 1

\Rightarrow 1 - b^{2} = - 1

\Rightarrow b^{2} = 2 \Rightarrow b = \pm 2

Hence

a = 1

and

b = \pm 2

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Question Text	If $a - b, a, a + b$ are zero of polynomial $x^{3} - 3 x^{2} + x + 1$ , then find $a$ and $b$ .
Answer Type	Text solution:1
Upvotes	150