Question

if alpha and beta are the zeros of a quadratic polynomial f(x)=3x2-7x-6, find a polynomial whose zeros are :i) alpha2 and beta2, ii)2alpha+3beta and 3alpha +2beta

Answer 1

Answer:

the polynomial obtained will be

x^2-(alpha+beta) x+ alpha beta

= x^2-35/3 x+3.

Step-by-step explanation:

Given that:

F(x) = 3x^2-7x-6

alpha and beta are the zeros of a quadratic polynomial

To find

find a polynomial whose zeros are

i) alpha2 and beta2

ii) 2alpha+3beta

iii) 3alpha +2beta

Solution:

alpha and beta are the zeros of a quadratic polynomial

Alpha + beta = -b/a

= 7/3

Alpha*beta = c/a

=-6/3 = -2

It is given that zeroes of polynomial are

i) alpha2 and beta2

ii) 2alpha+3beta

iii) 3alpha +2beta

So 2alpha+3beta+3alpha +2beta

= 5alpha + 5 beta

= 5(alpha+beta)

=5*(7/3) = 35/3

Now

(2alpha+3beta)(3alpha +2beta)

= 6alpha^2 + 13alphabeta + 6beta^2

= 6(alpha^2+beta^2) +13alphabeta

= 6[ (alpha+beta)^2 - 2 alphabeta]+ 13alphabeta

= 6[(7/3) ^2 -2(-2)]+13(-2)

= 6[49/9 +4 ] -26

= 29.4444-26

= 3.4444

Thus the polynomial obtained will be

x^2-(alpha+beta) x+ alpha beta

= x^2-35/3 x+3.4444

Answer 2

The answer of first point will be 6x2-49x-36

And second one check it on Google