Question

If alpha and beta are the zeros of the polynomial p(x)=2x^2-5x+7 find a polynomial whose zeros are are 2alpha + 3beta and 3alpha + 2beta.

Answer 1

Quadratic equation is of the form ax2+bx+c

now sum of roots=

-b

a

and product of roots=

c

a

here the eqn is 2x2-5x+7 and α and β are its roots.

sum of roots=α+β=

5

2

and product of roots=α×β=

7

2

now we have to form an equation whose roots are 2α+3β and 3α+2β.

equation will be of the form x2-sx+p......(1) where s and p represent sum and product of roots.

s=5(α+β)=

25

2

and p=6((α)2+(β)2)+13αβ)=6(α+β)2+αβ=

6

4

×25+

7

2

=41.

substituting s and p in (1) we get eqn=x2-

25

2

x+41=0.