Question

If alpha and beta are zeroes of p(x)=2x²+x+1 then find value of alpha⁴+beta⁴
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Answer 1

Answer:

by factorization method

2*1=2

2x^2+2x+x+1

2x(x+1)+1(x+1)

(x+1) (2x+1)=0

x+1=0. 2x+1=0

x=-1. x=-1/2

sum of zeroes=-b/a=-1/2

product of zeroes=c/a=1/2

alpha^4+beta^4

(1/2)^4

1/16

Answer 2

Answer:

Explanation:

Given :

• Polynomial, p(x) = 2x² + x + 1.
• α and β are the zeroes of polynomial.

To Find :

• The value of α⁴ + β⁴.

Solution :

Given ; polynomial, p(x) = 2x² + x + 1.

On comparing with, ax² + bx + c, we get ;

=> a = 2 , b = 1 , c = 1

Given ; α and β are the zeroes of polynomial.

Sum of zeroes = -b/a

=> α + β = -1/2

Product of zeroes = c/a

=> αβ = 1/2

Now,

α⁴ + β⁴ = [(α + β)² - 2αβ]² - 2(αβ)²

=> α⁴ + β⁴ = [(-1/2)² - 2 × (1/2)]² - 2(1/2)²

=> α⁴ + β⁴ = (1/4 - 1)² - 2 × 1/4

=> α⁴ + β⁴ = (-3/4)² - 1/2

=> α⁴ + β⁴ = 9/16 - 1/2

=> α⁴ + β⁴ = 1/16

Hence :

The value of α⁴ + β⁴ is 1/16.