Question

If α and β are the zeros of the polynomial x²-5x+6. Then find an quadratic polynomial whose roots are 2α and 2β.

Answer 1

Sum of zeroes is

2α + 2β

=2(α+β)

Product of zeroes

2α×2β

= 4αβ

__________________________

x²-5x+6

The sum of zeroes of this polynomial is

-(-5)/1 = 5

The product is

6/1 = 6

____________________________

A polynomial is given by

x² - (α+β) x + (αβ)

Hence the sum will be

2(α+β) = 2(5) = 10

Product

4(αβ) = 4(6) = 24

__________________________

Hence, the polynomial, will be

x² - 10x + 24

Answer 2

Solution :-

Given : α and β are the zeros of quadratic polynomial x² - 5x² + 6.

Roots are 2α and 2β.

In polynomial x² - 5x + 6,

α + β = - (-5)/1 = 5

αβ = 6/1 = 6

Sum of zeros = 2α + 2β

= 2(α + β)

Product of zeros = 2α × 2β

= 4αβ

By using quadratic formula,

x² - (α + β)x + αβ

= x² - 2(5)x + 4 × 6

∴ x² - 10x + 24

Hence,

Quadratic polynomial = x² - 10x + 24