Question

Rajesh donated some money and books to a school for poor children. Money and books can be represented by the zeroes (i.e alpha, beta) of the polynomial p(x) = 2x*x- 5x + 7. Akshitawho is friend of Rajesh, also got
inspired by him and donated the money and books in the form of a polynomial whose zeroes are 2alpha + 3beta and 3alpha + 2beta. Find the polynomial whose zeroes are 2alpha + 3beta and 3alpha+ 2beta.
Why did Akshitagot inspired by Rajesh?

Answer 1

α and β are the zeroes of 2x² - 5x + 7

α+β = -(-5)/2 = 5/2 ------------------(1)
αβ = 7/2 --------------------------------(2)

Akshita's donations are (2α+3β) and (3α+2β)
they are roots of a polynomial. let it be f(x) = ax² + bx + c

-b/a = (2α+3β) + (3α+2β)
       = 5α+5β = 5(α+β)
       = 5×(5/2)     (from eqn(1))
       = 25/2
c/a = (2α+3β) × (3α+2β)
      = 6α² + 13αβ + 6β²
      = 6α² + 12αβ + 6β² + αβ
      = 6(α²+β²+2αβ) + αβ
      = 6(α+β)² + αβ
      = 6(5/2)² + 7/2   (from eqn(1) and (2))
      = 6×25/4 + 7/2
      = 75/2 + 7/2
      = 82/2

Then f(x) = x² - (-b/a) x + (c/a) = 0
              ⇒  x² - (25/2) x + (82/2) = 0
              ⇒  x² - 25/2 x + 82/2 = 0
              ⇒ 1/2( 2x² - 25x + 82) = 0
              ⇒ 2x² - 25x + 82 = 0

The polynomial is 2x² - 25x + 82 = 0

Akshita was inspired by Rajesh because it was a act of charity and she was a good person who can also donate. Moreover, She found it interesting that the donation amounts are zeroes of another polynomial. ;-P