Question

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

Answer 1

Since α and β are the zeros of the quadratic polynomial f(x) = x² – 2x + 3

`alpha+beta="-coefficient of x"/("coefficient of "x^2)`

`=(-(-2))/1`

= 2

Product of the zeroes `="constant term"/("coefficient of "x^2)`

`=3/1`

= 3

Let S and P denote respectively the sums and product of the polynomial whose zeros

α + 2, β + 2

S = (α + 2) + (β + 2)

S = α + β + 2 + 2

S = 2 + 2 + 2

S = 6

P = (α + 2)(β + 2)

P = αβ + 2β + 2α + 4

P = αβ + 2(α + β) + 4

P = 3 + 2(2) + 4

P = 3 + 4 + 4

P = 11

Therefore the required polynomial f(x) is given by

f(x) = k(x² - Sx + P)

f(x) = k(x² - 6x + 11)

Hence, the required equation is f(x) = k(x² - 6x + 11)