Question

If alpha and beta are the zeros of the polynomial P(x) = x^2 – px + q then find the value of 1/α+1/β
Please show the steps.

Answer 1

$\alpha + \beta = - p$
$\alpha \times \beta = q$
$1 \div \alpha + 1 \div \beta = \alpha + \beta \div \alpha \times \beta$
by solving we get
answer
$- p \div q$

Answer 2

Hyyyy


Here is the answer

Let α and β are the zeros of the polynomial

Now, α + β = -p

=> (α + β)2 = (-p)2

=> (α + β)2 = p2

and α * β = q

Again (α - β)2 = (α + β)2 - 4 * α * β

=> (α - β)2 = (-p)2 - 4 * q

=> (α - β)2 = p2 - 4q

Now, the polynomial whose zeroes are (α + β)2 and (α - β)2 is

f(x) = x2 - {sum of zeros}x + product of zeros

=> f(x) = x2 - {(α + β)2 + (α - β)2 }x + (α + β)2 * (α - β)2

=> f(x) = x2 - {p2 + p2 - 4q}x + (p2 - 4q) * p2

=> f(x) = x2 - (2p2 - 4q)x + p4 - 4 * p2 * q

=> f(x) = x2 - (2p2 - 4q)x + p4 - 4 qp2

This is the required polynomial.

Hope helped