Question

Q3. If α and β are zeroes of a quadratic polynomial x2+4x+3, form the polynomial whose zeroes are 1+αβand1+βα.

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Answer 1

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Answer 2

Hi !

p(x) = x² + 4x + 3
a = 1
b = 4
c = 3

α and β are zeros of p(x)

sum of zeros = α + β = -b/a  = -4

product of zeros = αβ = c/a = 3

α + β = -4
αβ = 3
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given ,
 
1 + αβ and 1 +βα are zeros of a polynomial

sum of zeros = 1 + αβ +  1 +βα 
                     = 2 + 2αβ
                     = 2 + 2 × 3 
                     = 8
product of zeros = (1 + αβ ) (1 +βα )
                           = 1 + αβ + αβ + α²β²
                           = 1 + 2αβ + (αβ)²
                           = 1 + 2 × 3 + (3)²
                           = 1 + 6 + 9
                            = 16

a quadratic polynomial can be formed:-

x² - (sum of zeros)x + (product of zeros)
 
= x² - 8x + 16             ---> required polynomial