Question

If α and β ae the zeroes of the quadratic polynomial f (x) = x2 + x – 2, then find a polynomial whose zeroes are 2α + 1 and 2β + 1.

Answer 1

α and β ae the zeroes of the quadratic polynomial f (x) = x² + x – 2.

so, sum of zeroes = - coefficient of x/coefficient of x²

or, (α + β) = -1/1 = -1 ......(1)

and product of zeroes = constant/coefficient of x²

or, αβ = -2/1 = -2 .......(2)

now we have to find polynomial whose zeroes are 2α + 1 and 2β + 1

we know, any quadratic polynomial is in the form of, x² - (sum of zeroes)x + product of zeroes.

= x² - (2α + 1 + 2β + 1)x + (2α +1)(2β + 1)

= x² - {2(α + β) + 2}x + {4αβ + 2(α + β) + 1}

from equations (1) and (2),

= x² - {2(-1) + 2}x + {4(-2) + 2(-1) + 1}

= x² - 0.x + {-9}

= x² - 9

hence, required polynomial is x² - 9