Question

if alpha and beta are zeroes of the polynomial 2X-7x+5 then find a polynomial whose zeroes are 2alpha+beta And 2beta+3

Answer 1

Hi Mate !!


Given equation:- 2x² - 7x + 5 = 0

Let's factorise it by middle term splitting !!

2x² - 7x + 5 = 0

2x² - 2x - 5x + 5 = 0

2x ( x - 1 ) - 5 ( x - 1 ) = 0


( 2x - 5 ) ( x - 1 ) = 0

• ( 2x - 5 ) = 0

x = 5/2

• ( x - 1 ) = 0

x = 1


$so \: \: \alpha = \frac{5}{2} \: \: \:and \: \: \beta =1$

• Zeros of new polynomial are :-

$2 \alpha + \beta \: \: \: and \: \: 2 \beta + \alpha$
( ur question have an error the second Zeros will be 2beta + alpha )


$2 \alpha + \beta = 2 \times \frac{5}{2} + 1 = 5 +1 = 6$


$2 \beta + \alpha = 2 \times 1 + \frac{5}{2} = \frac{4 + 5}{2} = \frac{9}{2}$


• Sum of Zeros :-

$6 + \frac{9}{2} = \frac{12 + 9}{2} = \frac{21}{2}$


• Product of Zeros :-

$6 \times \frac{9}{2} = 27$

♯ To form the quadratic equation we have formula as :-

x² - ( Sum of Zeros )x + (Product of Zeros)

Putting value in it !!

${x}^{2} - \frac{21}{2} x + 27 = 0$


$\frac{2{x}^{2} - 21x +54 }{2} = 0$


2x² - 21x + 54 = 0 is the required quadratic equation !!