Question

If alpha and beta are the zeroes of the polynomial 3x2-5x+2 then what is the value of alpha3 + beta3?

Answer 1

let alpha and beta be m and n
m + n = 5/3
mn = 2/3

alpha^3 + beta^3
= m^3 + n^3
= (m+n)^3 - 3mn ( m+n )
= (5/3)^3 - 3×(2/3) (5/3)
= ( 125/27) - (10/3)
= ( 125 - 90 ) /27
= 35 / 27

Answer 2

Answer:

The value of $\alpha^3+\beta^3$ is found out to be 3 5/27

Explanation:

Let the given polynomial be $f(x) = 3x^2-5x+2$

For finding the zeroes or roots of any polynomial, f(x) should be equated to 0.

Thus the value will be,

$\begin{array} { c } { 3 x ^ { 2 } - 5 x + 2 = 0 } \\\\ { 3 x ^ { 2 } - 3 x - 2 x + 2 = 0 } \\\\ { 3 x ( x - 1 ) - 2 ( x - 1 ) = 0 } \\\\ { ( 3 x - 2 ) ( x - 1 ) = 0 } \\\\ { x = 1 \text { or } \frac { 2 } { 3 } } \end{array}$

Given alpha, beta are the roots of the given polynomial f(x),

i.e., alpha (α) = 1 , beta (β) = 2/3

Thus the value of  

$\begin{array} { c } { \alpha ^ { 3 } + \beta ^ { 3 } = ( 1 ) ^ { 3 } + \left( \frac { 2 } { 3 } \right) ^ { 3 } } \\\\ { = 1 + \frac { 8 } { 27 } } \\\\ { = \frac { 27 + 8 } { 27 } } \\\\ { = \frac { 35 } { 27 } } \end{array}$

Therefore, the value of $\alpha^3+\beta^3$ is found out to be 35/27.