Question

2x3 +x2 - 5x+2 ;1/2,1,-2 Verify the relationship between the coefficients and zeroes .​

Answer 1

Step-by-step explanation:

Given:-

2x^3 +x^2 - 5x+2 ;1/2,1,-2

To find:-

Verify the relationship between the coefficients and zeroes .?

Solution:-

Given cubic Polynomial

P(x)=2x^3 +x^2 - 5x+2

On Comparing this with the standard cubic Polynomial ax^3+bx^2+cx+d then

We have

a = 2

b=1

c=-5

d=2

and given zeroes are 1/2, 1, -2

Let they be

α = 1/2

β = 1

γ = -2

Relationship between the zeroes and coefficients:

i) Sum of the zeroes = α + β +γ

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

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

=(1/2)+(-1)

=(1/2)-1

=(1-2)/2

=-1/2

α + β +γ = -1/2

-b/a

= -1/2

α + β +γ = -b/a

ii) Sum of the product of the two zeroes taken at a time = α β + β γ +γ α

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

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

=(1/2)-2-1

=(1/2)-3

=(1-6)/2

α β + β γ +γ α =-5/2

c/a

= -5/2

α β + β γ +γ α = c/a

iii) Product of the zeroes= α β γ

=(1/2)(1)(-2)

=-2/2

α β γ =-1

-d/a

=-(2/2)

=-1

α β γ = -1

Verified the given relationship between the zeroes and coefficients of the given cubic polynomial.

Used formulae:-

• The standard cubic Polynomial ax^3+bx^2+cx+d .

• Sum of the zeroes = α + β +γ = -b/a

• Sum of the product of the two zeroes taken at a time = α β + β γ +γ α = c/a

• Product of the zeroes= α β γ =-d/a