(Q24) Find the quadratic polynomial,for the zeros α and β given in each case

(i)2,-1   (ii)√3,-√3  (iii)
1
4
 , -1  (iv)
1
2
 ,
3
2

Answer:

(i)Let the polynomial be ax2+bx+c,a≠0 and its zeros α and β

Here α=2 and β= -1

Sum of the zeros =α+β=+(-)=

product of the zeros=αβ=×(-)= -

therefore the quadratic polynomial ax2+bx+c is x2-(α+β)x+αβ

the quadratic polynomial will be x2-x-

(ii)Let the zeros be α=√3 and β= -√3

sum of the zeros =α+β

=√+(-√)=

product of the zeros αβ

=√×(-√)= -

the quadratic polynomial

ax2+bx+c is[x2-(α+β)x+αβ]

=[x2-.x+(-)]

the quadratic polynomial will be x2-

(iii)Let the zeros be α=
1
4
 and β=-1

sum of the zeros =α+β

=
+(-) =
-
4

product of the zeros=αβ

=
×(-)=
-
4

the quadratic polynomial ax2+bx+c is [x2-(α+β)x+αβ]

=[x2-(
-
4
)x+(
-
4
)]

the quadratic polynomial will be 4x2+x-

(iv)Let the zeros be α=
1
2
 and β=
3
2

sum of the zeros =α+β

=
+
=

product of the zeros=αβ

=
×
=
4

the quadratic polynomial =ax2+bx+c is[x2-(α+β)x+αβ]

=[x2-x+(
4
)]

The quadratic polynomial will be 4x2-x+