(Q1) Verify that the numbers given along side the cubic polynomial below are their zeros.Also verify the relationship between the zeros and the coeffcients in each case
2x3+x2-5x+2 (
1
2
,1, -2)

Answer

Given polynomial 2x3+x2-5x+2 comparing the given polynomial with ax3+bx2cx+d

We get a=, b= c= - and d=

p(
1
2
) = 2(
1
2
)3+(
1
2
)2 - 5(
1
2
)+2
=
1
4
+
1
4
 -
5
2
+
2
1

=
4
 =

p(1)=2(1)3+(1)2 -5(1) +2

=2+1-5+2=

p(-2)=2(-2)3+(-2)2 - 5(-2)+2

=2(-8)+4+10+2

=

1
2
,1 and -2 are the zeros of 2x3+x2 - 5x+2
So, α=
1
2
, β=1 and γ= -2 therefore

α+β+γ=
1
2
+1+(-2)

=
-
 =
-b
a

αβ+βγ+γα= (
1
2
)(1)+(1)(-2)+(-2)(
1
2
)

=
1
2
 - 2 - 1=
-
2
 =
c
a

αβγ=
1
2
×1×(-2)= -=
-
2
 =
-d
a