(Q11)Verify whether 3,-1 and -
1
3
 are the zeros of the cubic polynomial p(x)=3x3-5x2-11x-3,and then verify the relationship between the zeros and the coefficients.

Answer:

p(x)=3x3-5x2-11x-3 is the given polynomial

Then p(3)=3×33-(5×32)-(11×3)-3=

p(-1)=3×(-1)3-5×(-1)2-11×(-1)-3=

p( -
1
3
) = 3×( -
1
3
) 3 - 5×( -
1
3
)2 - 11×( -
1
3
) - 3
= -
1
9
 -
5
9
 +
11
3
 - 3=
Therefore 3,-1 and -
1
3
 are the zeros of 3x3-5x2-11x-3
So,we take α=3,β= -1 and γ= -
1
3

Comparing the given polynomials with ax3+bx2+cx+d,we get a=3,b= -5,c= -11,d= -3

Now,

α+β+γ=3+(-1)+( -
1
3
) = 2-
1
3
 =
=
-(-)
 =
-b
a
αβ+βγ+γα=3×(-1)+(-1)×( -
1
3
) + ( -
1
3
)×3= -3+
1
3
 - 1=
-
=
c
a
,
αβγ=3×(-1)×( -
1
3
)==
-(-)
3
 =
-d
a