Find the zeroes of the cubic polynomial x3 +6x2 +11x +6 and verify the realtionship between the zeroes and the coefficients.
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Step-by-step explanation:
Let p(x)=x
3
−6x
2
+11x−6
Then, p(1)=(1)
3
−6(1)
2
+11(1)−6
=1−6+11−6
=0
p(2)=(2)
3
−6(2)
2
+11(2)−6
=8−24+22−6
=0
p(3)=(3)
3
−6(3)
2
+11(3)−6
=27−54+33−6
=0
Hence, 1,2 and 3 are the zeroes of the given polynomial
x
3
−6x
2
+11x−6.
Now, Let α=1,β=2 and γ=3
Then, α+β+γ=1+2+3=6
=−
Coefficientofx
3
Coefficientofx
2
=−
1
−6
=6
αβ+βγ+γα=(1)(2)+(2)(3)+(3)(1)
=2+6+3
=11
=
Coefficientofx
3
Coefficientofx
=
1
11
=11
And αβγ=1×2×3
=6
=−
Coefficientofx
3
Constantterm
=−
1
−6
=6
Thus, the relationship between the zeroes and the coefficients is verified.
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