Question

If α and β are the zeros of the quadratic polynomial f(x) = x² - 2x + 3 , find a polynomial whose roots are (i) α + 2, β + 2 (ii) (α - 1)/(α + 1), (β - 1)/(β + 1) .

Answer 1

HERE IS YOUR ANSWER

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MARK BRAINLIEST ⭐

Answer 2

(i)  f(x) = k(x² -9x +11)

(ii)  f(x) = k(x² -2/3x +1/3)

Step-by-step explanation:

 f(x) = x² - 2x + 3

α and β are the zeroes of the polynomial.

Then α + β = -b/a = -(-2)/1 = 2

αβ = c/a  = 3/1 = 3

 

(i) α + 2, β + 2 are the zeroes.

Then the polynomial will be f(x) = k(x² -Sx +P)

S =  α + 2 + β + 2 = 4 + (α + β) = 4 + 2 + 3 = 9

P = (α + 2)(β + 2) = (αβ + 2((α + β) + 4) = 3 + 2(2) + 4 = 11

So the polynomial will be:

f(x) = k(x² -9x +11)

(ii) (α - 1)/(α + 1), (β - 1)/(β + 1).

S = (α - 1)/(α + 1) + (β - 1)/(β + 1).

  = [(α - 1)(β + 1) + (α + 1)(β - 1) ]/ (α + 1)(β + 1)

  = (αβ + α-β - 1) + (αβ - α +β - 1) / (αβ + α +β + 1)

  = (2αβ - 2) / (αβ + α+β + 1)

  = 2(3) - 2  / 3 + 2 + 1

  = 4/6

  = 2/3

P = (α - 1)/(α + 1) * (β - 1)/(β + 1).

  =  (α - 1)(β - 1) / (α + 1)(β + 1)

  = (αβ - α-β + 1) / (αβ + α +β +1)

  = (3 - 2 + 1) / (3 + 2 + 1)

  = 2/6

  = 1/3

So the polynomial will be:

f(x) = k(x² -2/3x +1/3)