Question

if alpha and bita are the two zeroes of the polynomial x2-5x+4 find the quadratic polynomial having zeroes alpha+1/bita and bita+1/alpha​

Answer 1

Explanation:

Given that α & β are zero of polynomial

f(x)=x

2

−3x−2

therefore α+β=3

αβ=−2

Now, the zero of the required quadratic polynomial are,

2α+β

1

&

2β+α

1

Sum of the roots-

2α+β

1

+

2β+α

1

=

(2α+β)(2β+α)

2β+α+2α+β

=

4αβ+2α

2

+2β

2

+αβ

3(α+β)

=

4×(−2)+2[(α+β)

2

−2αβ]+(−2)

3×3

=

−10+2[9+2×2]

9

=

−10+26

9

=

16

9

Products of roots:-

2α+β

1

×

2β+α

1

=

4αβ+2[(α+β)

2

−2αβ]+αβ

1

=

16

1

Now Req eq.

x

2

−(sum of roots)x+ Product of roots=0

=x

2

−

16

9

x+

16

1

=0

=16x

2

−9x+16=0.