Question

if α and β are zeroes of polynomial of the polynomial x²+x-2 find 1/α - 1β
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Answer 1

Step-by-step explanation:

Solution

Answer:

Given :-

polynomial , x² + x - 2 = 0

α and β are zeroes of this polynomial

Find :-

Value of 1/α - 1/β

Explanation

Using Formula,

★ Sum of zeroes = -(coefficient of x)/(coefficient of x²)

★product of zeroes = (constant part)/(coefficient of x²)

★ ( α - β) = √[(α + β)² - 4α β]

________________________

Now, calculate,

➡ Sum of zeroes = -(coefficient of x)/(coefficient of x²)

➡ (α + β) = -1 __________(1)

Again,

➡product of zeroes = (constant part)/(coefficient of x²)

➡ (α β) = -2 _____________(2)

So, Now calculate (α - β)

➡ ( α - β) = √[(α + β)² - 4α β]

keep all above values

➡ ( α - β) = √[(-1)² - 4*(-2)]

➡( α - β) = √(1+8)

➡( α - β) = √9

➡( α - β) = 3 _______________(3)

Now, calculate (1/α - 1/β)

➡ (1/α - 1/β) = ( α - β)/α β

Keep value by equ(2) & equ(3)

➡ (1/α - 1/β) = 3/(-2)

➡ (1/α - 1/β) = -3/2

Hence

Value of (1/α - 1/β) be = -3/2

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Answer 2

Zeros of polynomial are alpha - beta , alpha , alpha + beta 

Sum of zeros =  (alpha - beta) + (alpha) + (alpha + beta) =  - b / a

                                       

3 * alpha    =   3 / 1 

                                               

alpha  =  1

product of zeros = (alpha - beta) *(alpha) *(alpha + beta)  =  - d /a

Now put value of alpha = 1

                   

(1 - beta) * 1 * (1+ beta)  =  -1

                                             

12 -  beta2   =  -1

                                             

- beta2 =  -1-1

                                             

beta 2= 2

                                               

beta =  + root 2

So alpha = 1   ,    beta = + root 2

alpha + beta = 1 + root 2

Thanks