If $α, β$ and $γ$ are the roots of the equation $x^{3} - 3 x^{2} + 5 x - 9 = 0$ then the value of the expression

$(α + β - γ) (β + γ - α) (γ + α - β)$ .

__

Open in App

Solution

Solution: Given equation $x^{3} - 3 x^{2} + 5 x - 9 = 0$

$α + β + γ = 3$ or $\sum α = 3$

$α β + β γ + γ α = 5$

$α β γ = 9$

$(α + β - γ) (β + γ - α) (γ + α - β)$

$\Rightarrow (\sum α - 2 γ) (\sum α - 2 α) (\sum α - 2 β)$

$(3 - 2 γ) (3 - 2 α) (3 - 2 β)$

$= (9 - 6 α - 6 γ + 4 α γ) (3 - 2 β)$

$= (9 - 6 α - 6 γ + 4 α γ) (3 - 2 β)$

$= (27 - 18 β - 18 α + 12 α β - 18 γ + 12 γ β + 12 α γ - 8 α β γ$

$= 27 - 18 \times 3 + 12 \times 5 - 8 \times 9$

$= 27 - 54 + 60 - 72$

$= - 39$

Suggest Corrections

Similar questions

If α, β and γ are the roots of the equation $x^{3}$ - 3 $x^{2}$ + 5x - 9 = 0 then the value of the expression

( α + β - γ) ( β + γ -α) (γ + α - β).

Q. If

α, β, γ

are the roots of

a x^{3} + b x^{2} + c x + d = 0

and

∣ ∣ ∣ ∣ ∣ \begin{matrix} α & β & γ β & γ & α γ & α & β \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

α \neq β \neq γ

, then find the equation whose roots are

α + β - γ, γ + α - β, β + γ - α

Q. If

α, β, γ

are the roots of

p x^{3} + q x^{2} + r = 0,

then the value of the determinant

∣ ∣ ∣ ∣ ∣ \begin{matrix} α β & β γ & γ α β γ & γ α & α β γ α & α β & β γ \end{matrix} ∣ ∣ ∣ ∣ ∣

Q. If

{2 x}^{3} + {3 x}^{2} + 5 x + 6 = 0

has roots

α, β, γ

then find

α + β + γ, α, β + β γ + γ α

and

α β γ

If $α, β a n d γ$ are the roots of the equation $x^{3} + 3 x + 2 = 0$ , Find the equation whose roots are $(α - β) (α - β), (β - γ) (β - α), (γ - α) (γ - β)$ .

Join BYJU'S Learning Program

If α,β and γ are the roots of the equation x3−3x2+5x−9=0 then the value of the expression (α+β−γ)(β+γ−α)(γ+α−β). __

If $α, β$ and $γ$ are the roots of the equation $x^{3} - 3 x^{2} + 5 x - 9 = 0$ then the value of the expression

$(α + β - γ) (β + γ - α) (γ + α - β)$ .

__