If-are-the-three-roots-of-the-3x-3-12x-2-77x-11-0-find-the-value-of-1-1-1-

Question Number 119408 by ZiYangLee last updated on 24/Oct/20

If α, β, γ are the three roots of the

3 x^{3} + 12 x^{2} - 77 x + 11 = 0,

find the value of (α - 1) (β - 1) (γ - 1) .

Commented by liberty last updated on 24/Oct/20

l e t p (x) = 3 x^{3} + 12 x^{2} - 77 x + 11 a n d

p (x) = 0 h a s t h e r o o t s a r e α, β a n d γ

l e t p (x + 1) = 3 {(x + 1)}^{3} + 12 {(x + 1)}^{2} - 77 (x + 1) + 11

h a s t h e r o o t s a r e {\begin{cases} α - 1 \\ β - 1 \\ γ - 1 \end{cases}

p (x + 1) = 3 (x^{3} + 3 x^{2} + 3 x + 1) + 12 (x^{2} + 2 x + 1) - 77 x - 77 + 11

p (X) = 3 x^{3} + 21 x^{2} - 44 x - 51

t h u s (α - 1) (β - 1) (γ - 1) = \frac{51}{3} = 17

Answered by mr W last updated on 24/Oct/20

α + β + γ = - \frac{12}{3}

α β + β γ + γ α = - \frac{77}{3}

α β γ = - \frac{11}{3}

(α - 1) (β - 1) (γ - 1)

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

= - \frac{11}{3} - \frac{12}{3} + \frac{77}{3} - 1

= 17

Commented by ZiYangLee last updated on 24/Oct/20

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