Question

Let α, β be the zeroes of polynomial p(x) = x2 -2x +7. Find a quadratic polynomial whose
zeroes are 2α and 2β

Answer 1

Let α, β be the zeroes of polynomial

p(x) = x2 -2x +7

Find relationship between zeroes :

Sum of zeroes = -b/a

α + β = -(-2)/1

α + β = 2

Product of zeroes = c/a

α × β = 7/1

α × β = 7

Here we have to find a polynomial whose zeroes are 2α and 2β.

Sum of zeroes = 2α + 2β

Sum of zeroes = 2(α + β)

Sum of zeroes = 2 × 2

Sum of zeroes = 4

Product of zeroes = 2α × 2β

Product of zeroes = 4αβ

Product of zeroes = 4 × 7

Product of zeroes = 28

Find the polynomial with the given zeroes :

f(x) = x² - (α + β)x + (αβ)

f(x) = x² - (4)x + (28)

f(x) = x² - 4x + 28

Therefore the required polynomial is

x² - 4x + 28