Step-by-step explanation:

♦ Given ♦ :-

• α, β, γ are the roots of x³-x²+12x-3 = 0

♦ To find ♦ :-

• The value of (1/α )+(1/β)+(1/γ).

♦ Solution ♦ :-

Given Cubic equation is x³-x²+12x-3 = 0

On comparing with the standard cubic equation ax³+bx²+cx+d = 0 then

a = 1

b = -1

c = 12

d = -3

Given roots of the equation = α, β, γ

We know that

Sum of the roots = -b / a

=> α + β + γ = -(-1)/1

=> α + β + γ = 1/1

=> α + β + γ = 1 -------------------(1)

Sum of the product of the two roots taken at a time = c/a

=> α β + γ β + γα = 12/1

=> α β + γ β + γα = 12 ----------(2)

and

Product of the roots = -d/a

=> α β γ = -d/a

=> α β γ = -(-3)/1

=> α β γ = 3/1

=> α β γ = 3 -----------------------(3)

Now, The value of (1/α )+(1/β)+(1/γ)

= (γ β + γα + α β)/(α β γ)

= (α β + γ β + γα)/(α β γ)

= 12/3

[from (3)&(4)]

= 4

♦ Answer ♦ :-

• The value of (1/α )+(1/β)+(1/γ) = 4

♦ Points To Know ♦:-

• The standard cubic equation is ax³+bx²+cx+d = 0.
• A cubic equation has at most three roots .
• Sum of the roots = -b/a
• Sum of the product of the two roots taken at a time = c/a
• Product of the roots = -d/a

Answer:

4

Step-by-step explanation:

Given Equation : [tex]x {}^{3} - x {}^{2} + 12x - 3[/tex]

and, [tex] \alpha \: , \beta \: \: and \: \: \gamma[/tex] are the roots of given equation.

so, SUM OF ROOTS = [tex] \frac{ - b}{a} [/tex]

[tex] \alpha + \beta + \gamma = \frac{ - b}{a} [/tex]

[tex] \alpha + \beta + \gamma = \frac{ - ( - 1)}{1} [/tex]

[tex] \alpha + \beta + \gamma = 1 \: \: - - - (i)[/tex]

and, PRODUCT OF ROOTS = [tex] \frac{ - d}{a} [/tex]

[tex] \alpha \beta \gamma = \frac{ - d}{a} [/tex]

[tex] \alpha \beta \gamma = \frac{ - ( - 3)}{1} [/tex]

[tex] \alpha \beta \gamma = 3 \: \: - - - - (ii)[/tex]

also, SUM OF THE PRODUCTS OF TWO ROOT = [tex] \frac{c}{a} [/tex]

[tex] \alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a} [/tex]

[tex] \alpha \beta + \gamma \beta + \gamma \alpha = \frac{12}{1} [/tex]

[tex] \alpha \beta + \gamma \beta + \gamma \alpha = 12[/tex] ---- (iii)

NOW, [tex] \frac{1}{ \alpha } + \frac{1}{ \beta } + \frac{1}{ \gamma } [/tex]

[tex] \frac{ \alpha \beta + \gamma \beta + \gamma \alpha }{ \alpha \beta \gamma } [/tex]

[tex] \frac{12}{3} [/tex]

from 3rd and 2nd equation

[tex]4[/tex]