Question

if α and β are the two zeroes of polynomial x2-2x+5 ,find a quadratic polynomial whose zeroes are α+β and 1/α+1/β.

Answer 1

HI there !

p(x) = x² -2x + 5

α and β are zeros of p(x)

α + β = -b/a = 2

αβ = 5


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Zeros of other polynomial :-

α + β = 2

1/α + 1/β = α + β /αβ
= 2/5


here ,

2 and 2/5 are zeros of a polynomial

sum of zeros = 2 + 2/5 = 10/5 + 2/5 = 12/5

product of zeros = 2×2 /5 = 4/5

a quadratic polynomial is given by:-

k [ x² - (sum of zeros )x + (product of zeros) ]

k [ x² -12/5x + 4/5 ]

5[x² - 12/5x + 4/5]

5x² -12x + 4 -----> required polynomial

Answer 2

Answer:

p(x) = x² - 2x + 5

Here, a= 1, b= -2, c= 5

Sum of zeros- α+β = -b/a = 2

Product of zeros- αβ = c/a = 5

Now,

Zeros of other polynomial, are α+β and 1/α + 1/β

So,

Sum of zeros = α +β +1/α +1/β = -b/a

= (α+β) + (α+β / αβ)

= (2) + (2/5)

= 12/5

Product of zeros = (α+β)(1/α + 1/β) = c/a

= (α+β)(α+β / αβ)

= 2 × 2/5

= 4/5

Hence, for the new equation,

a= 5 , b= -12 , c= 4

Putting these values in ax² + bx + c

= (5)x² + (-12)x + (4)

= 5x² -12x +4

      OR

(dividing the equation by 5)

= x² - 12x/5 + 4/5

Both the answers are considered correct.