Question

If alpha and beta are the zeroes of x²-x-2 ,form a quadratic polynomial whose zeroes are. 2 alpha +1, 2beta +1

Answer 1

Answer:

The required polynomial is

$x^2-4x-5$

Step-by-step explanation:

Concept:

If \alpha and \beta are roots of $ax^2+bx+c=0$ then

$sum\: of \:roots=\alpha+\beta=\frac{-b}{a}\\\\product\: of\: roots=\alpha.\beta=\frac{c}{a}$

$Given:\\\\\alpha\:and\:\beta \:are\:zeros\:of x^2-x-2$

Then,

$\alpha+\beta=1\\\\\alpha.\beta=-2$

sum of the roots

$=(2\alpha+1)+(2\beta+1)\\\\=2\alpha+2\beta+2\\\\=2(\alpha+\beta+1)\\\\=2(1+1)\\\\=4$

product of the roots

$=(2\alpha+1)(2\beta+1)\\\\=4\alpha\beta+2(\alpha+\beta)+1\\\\=4(-2)+2(1)+1\\\\=-8+3\\\\=-5$

The required polynomial is

$x^2-(sum\;of\:roots)x+(product\:of\:roots)\\\\=x^2-(4)x+(-5)\\\\=x^2-4x-5$

Answer 2

Answer:

Step-by-step explanation:

if alpha and beta are zeros then

alpha +beta=1

alpha*beta=-2

sum of zeros=(2 alpha+1)+(2 beta+1)

2 alpha+2 beta+2

2(alpha+beta+1)

2(1+1)

2*2=4

product of zeros=(2 alpha+1)*(2 beta+1)

4 alpha*beta+2(alpha +beta)+1

4(-2)+2(1)+1

-8+3

-5

x^2-(sum of zeros)x+product of  zeros

x^2-4x-5