Question

if the zeroes of polynomial x³-3x²+x+1 are a-b , a , a+b then find a and b
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Answer 1

Step-by-step explanation:

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Answer 2

zero of a polynomial is a value of the variable at which the polynomial equals 0 .

For the said polynomial, we have these equations:

a3−3a2+a+1=0

(a−b)3−3(a−b)2+a−b+1=0

or,a3−3a2b+3ab2−b3−3a2+6ab−3b2+a−b+1=0

3.(a+b)3−3(a+b)2+a+b+1=0

or,a3+3a2b+3ab2+b3−3a2−6ab−3b2+a+b+1=0

Using equations 1 & 2, we get,

−3a2b+3ab2−b3−3b2+6ab−b=0

or, 3a2b−3ab2+b3+3b2−6ab+b=0

Using equations 1 & 3, we get,

3a2b+3ab2+b3−3b2−6ab+b=0

Subtracting last two equations, we get,

6ab2−6b2=0

or, 6b2(a−1)=0

or, b=0, or a=1

Now, putting a=1 in either equation 1 or equation 2, we get,

3b+3b2+b3−3b2−6b+b=0

or, b3−2b=0

or, b(b2−2)=0

or, b=0 , or b=2–√, or b=−2–√

So, we have got a=1 and b=0,2–√,−2–√

Therefore, the zeroes of the said polynomial are 1,1+2–√ and 1−2–√.