Question

if alpha and beta are zeroes of x2-3x+2 find alpha3 +beta3​

Answer 1

Answer

9

$\rule{200}2$

Explanation

Given,

$\sf\alpha \ and \beta$ are zeroes of polynomial x² - 3x + 2.

To find,

$\sf {\alpha}^3 + {\beta}^3$

We know that in a quadratic polynomial, sum of zeroes = -b/a

and product of zeroes = c/a

Comparing the given polynomial with standard quadratic polynomial ax² + bx + c, we get

a = 1

b = -3

c = 2

So, we get

$\sf \alpha + \beta = \frac{- (-3)}{1}$

$\sf \alpha + \beta = 3$ ........(1)

And, similarly

$\sf \alpha\beta = 2$............(2)

Now, we know that

(x + y)³ = x³ + y³ + 3xy(x + y)

→ x³ + y³ = (x + y)³ - 3xy(x + y)

So,

$\sf \alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$

From (1) and (2),

$\sf \alpha^3 + \beta^3 = (3)^3 - 3(2)(3)$

$\sf \alpha^3 + \beta^3 = 27 - 18$

$\sf \alpha^3 + \beta^3 = 9$