Question

if alpha and beta are the zeroes of the polynomial 2x²-4x+5,then find the value of
${ \alpha }^{3} + { \beta }^{3}$
​

Answer 1

If α and β are Zeroes of a Quadratic Polynomial, ax² + bx + c then :

●  α and β are roots of the Quadratic Equation, ax² + bx + c = 0

●  $\mathsf{Sum\;of\;the\;Roots\;(\alpha + \beta) = \dfrac{-b}{a}}$

●  $\mathsf{Product\;of\;the\;Roots\;(\alpha.\beta) = \dfrac{c}{a}}$

Given : α and β are zeros of Quadratic Polynomial 2x² - 4x + 5

Here : a = 2 and b = -4 and c = 5

●  α and β are roots of the Quadratic Equation, 2x² - 4x + 5 = 0

●  $\mathsf{Sum\;of\;the\;Roots\;(\alpha + \beta) = \dfrac{4}{2} = 2}$

●  $\mathsf{Product\;of\;the\;Roots\;(\alpha.\beta) = \dfrac{5}{2}}$

Consider : (α + β)³

$:\implies$  (α + β)³ = α³ + 3α²β + 3αβ² + β³

$:\implies$  (α + β)³ = α³ + 3αβ(α + β) + β³

$:\implies$  α³ + β³ = (α + β)³ - 3αβ(α + β)

$\mathsf{\implies \alpha^3 + \beta^3 = 2^3 - 3\bigg(\dfrac{5}{2}\bigg)(2)}$

$\mathsf{\implies \alpha^3 + \beta^3 = 8 - (3\times 5)}$

$\mathsf{\implies \alpha^3 + \beta^3 = 8 - 15}$

$\mathsf{\implies \alpha^3 + \beta^3 = -7}$

Answer 2

alpha+beta=-b/a

alpha+beta=-(-4/2)

alpha+beta=2

alpha*beta=c/a

alpha*beta=5/2

now using identity

alpha^3+beta^3=(alpha+beta)^3-3alpha*beta(alpha+beta)

now putting up the values we get,

(2)^3-3*5/2(2)

8-15

-7 is the answer