Question

If α and β be the two zeroes of the quadratic polynomial P(x) = 2x2 – 3x + 7. Evaluate

a) α2 + β2 b) α3 + β3 c ) 1/ α+ 1/β

Answer 1

see explanation

Explanation:

Given

p(x) = 2x² - 3x + 7

with a = 2, b = - 3, c = 7

If α and β are the zeros, then

the sum of the roots

α + β = -
`(-3)/(2)` =
`(3)/(2)` ←

and the product of the roots

αβ =
`(c)/(a)` =
`(7)/(2)` ←

(a)

α² + β² = (α + β)² - 2αβ = (
`(3)/(2)` )² - 2 ×
`(7)/(2)` =
`(9)/(4)` - 7 = -
`(19)/(4)`

------------------------------------------------------------------------------------------

(b)

α³ + β³

= (α + β)³ - 3α²β - 3αβ²

= (α + β)³ - 3α(α + β)

= (
`(3)/(2)` )³ - 3 ×
`(7)/(2)` ×
`(3)/(2)`

=
`(27)/(8)` -
`(63)/(4)` = -
`(99)/(8)`

-----------------------------------------------------------------------------------------------

(c)

`(1)/(\alpha )` +
`(1)/(\beta )`

=
`(\alpha+\beta  )/(\alpha\beta  )`

=
`(3)/(2)` ×
`(2)/(7)` =
`(3)/(7)`