Question

Verify whether 1,2aand 3 are the zeroes of the cubic polynomial p(x)=x square -6 x square + 11x-6 and then check the relation ship between zeroes and the co efficient

Answer 1

Answer:

Step-by-step explanation:

Let p (x) = x ³ - 6x² + 11x - 6

If 1 is the zero of the polynomial

p ( 1 ) = ( 1 )³ - 6 ( 1 )² + 11 ( 1 )- 6

= 1 - 6 ( 1 ) + 11 - 6

= 1 - 6 + 11 - 6

= 12 - 12

= 0

∴ 1 is the zero of the polynomial of p (x).

If 2 is a zero of the polynomial

p ( 2 ) = ( 2 )³ - 6 ( 2 )² + 11 ( 2 ) - 6

= 8 - 6 ( 4 ) + 22 - 6

= 8 - 24 + 22 - 6

= 30 - 30

= 0

∴ 2 is also the zero of the polynomial of p (x).

If 3 is a zero of the polynomial

p ( 3 ) = ( 3 )³ - 6 ( 3 )² + 11 ( 3 ) - 6

= 27 - 6 ( 9 ) + 33 - 6

= 27 - 54 + 33 - 6

= 60 - 60

= 0

∴ 3 is also the zero of the polynomial of p ( x).

Checking :

Comparing the given equation with ax³ + bx² + cx + d ,we have

a = 1 , b = - 6 , c = 11 and d = - 6. And α = 1 , β = 2 , γ = 3

α + β + γ = - b / a

1 + 2 + 3 = - ( -6 ) / 1

6 = 6 / 1

6 = 6.

αβ + βγ + γα = c / a

1 ( 2 ) + 2 ( 3 ) + 3 ( 1 ) = 11 / 1

2 + 6 + 3 = 11

11 = 11.

αβγ = - d / a

1 × 2 × 3 = - ( - 6 ) / 1

6 = 6/ 1

6= 6.