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If the zeroes of the polynomial x^{3} - 3x^{2}+ x+1 are a - b

Question

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

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Daniel Discussed

If the zeroes of the polynomial

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

are

a - b, a, a + b

, find

a

and

b .

12 mins ago

Discuss this question LIVE

12 mins ago

Text solutionVerified

Polynomial

p (x) = x^{3} - 3 x^{2} + x + 1

a - b, a, a + b

are zeroes of the polynomial

p (x) = x^{3} - 3 x^{2} + x + 1

∴

Sum of its zeroes

= (a - b) + a + (a - b)

= - \frac{- 3}{1}

\Rightarrow 3 a = 3 \Rightarrow a = 1

Sum of the product of its zeroes

= a (a - b) + a (a + b) + (a - b) (a + b) = \frac{1}{1}

\Rightarrow a^{2} - ab + a^{2} + ab + a^{2} - b^{2} = 1

\Rightarrow 3 a^{2} - b^{2} = 1 \Rightarrow 3 (1)^{2} - b^{2} = 1

\Rightarrow b^{2} = 3 - 1 = 2 \Rightarrow b = \pm 2

Since

a = 1

and

b = \pm 2

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