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Question Number 39827 by Rio Mike last updated on 11/Jul/18

i f f (x) = 3 x^{3} + {p x}^{2} + 4 x - 8

a n d (x - 1) i s a f a c t o r o f

f (x) .

a) f i n d t h e v a l u e o f p .

w i t h t h e s e v a l u e o f p

b) s o l v e t h e e q u a t i o n f (x) = 0 .

i f α a n d β a r e r o o t s o f

f (x),

c) f i n d α + β a n d α β

d) E v a l u a t e α^{2} + β^{2} h e n c e α - β

Answered by MJS last updated on 11/Jul/18

3 (x - 1) (x - α) (x - β) = 3 x^{3} + {p x}^{2} + 4 x - 8

3 x^{3} - 3 (α + β + 1) x^{2} + 3 (α + α β + β) x - 3 α β =

= 3 x^{3} + {p x}^{2} + 4 x - 8

\Rightarrow

p = - 3 (α + β + 1)

3 (α + α β + β) = 4

3 α β = 8 \Rightarrow α = \frac{8}{3 β}

3 (\frac{8}{3 β} + \frac{8}{3 β} β + β) = 4 \Rightarrow β^{2} + \frac{4}{3} β + \frac{8}{3} = 0 \Rightarrow

\Rightarrow β = - \frac{2}{3} \pm \frac{2 \sqrt{5}}{3} i

\Rightarrow α = - \frac{2}{3} \mp \frac{2 \sqrt{5}}{3} i

\Rightarrow p = 1

α + β = - \frac{4}{3}

α β = \frac{8}{3}

α^{2} + β^{2} = - \frac{32}{9}

α - β = \pm \frac{4 \sqrt{5}}{3} i

Commented by Rio Mike last updated on 12/Jul/18

t h e c u b i c e x p r e s s i o n i s o f

t h e f o r m

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

s o w h a t i s α + \overset{}{β} = ?

a n d α β = ?

Commented by MJS last updated on 12/Jul/18

α, β are the roots besides 1 which is given in

this case .

{a x}^{3} + {b x}^{2} + c x + d = a (x - α) (x - β) (x - γ) =

= a (x^{3} - (α + β + γ) x^{2} + (α β + α γ + β γ) x - α β γ)

in our case γ = 1 leads to

a (x^{3} - (α + β + 1) x^{2} + (α + α β + β) x - α β)

Answered by Rasheed.Sindhi last updated on 13/Jul/18

AnOther approach

If f (x) is divided by x - 1 then the

quotient Q (x) and remainder R can

be determined by synthetic division :

(\begin{matrix} 1) 3 & p & 4 & - 8 \\ 3 & p + 3 & p + 7 \\ 3 & p + 3 & p + 7 & ∣ \overset{―}{p - 1} \end{matrix})

∵ x - 1 is factor of f (x)

∴ R = p - 1 = 0 \Rightarrow p = 1 (a)

Q (x) = {3 x}^{2} + (p + 3) x + (p + 7)

= {3 x}^{2} + (1 + 3) x + (1 + 7)

= {3 x}^{2} + 4 x + 8

f (x) = (x - 1) . Q (x) + R

[Relation between divedend, divisor & remainder]

= (x - 1) . Q (x) + 0

= (x - 1) ({3 x}^{2} + 4 x + 8)

(b) f (x) = 0 \Rightarrow x - 1 = 0 ∣ {3 x}^{2} + 4 x + 8 = 0

x = 1 ∣ x = \frac{- 4 \pm \sqrt{16 - 96}}{6}

∣ x = \frac{- 2 \pm 2 i \sqrt{5}}{3}

Let α & β are the roots of {3 x}^{2} + 4 x + 8 = 0

α + β = - \frac{4}{3} & α β = \frac{8}{3}

α - β = \frac{- 2 + 2 i \sqrt{5}}{3} - \frac{- 2 - 2 i \sqrt{5}}{3}

= \frac{4 \sqrt{5}}{3} i

α^{2} + β^{2} = {(α + β)}^{2} - 2 α β = {(- \frac{4}{3})}^{2} - 2 (\frac{8}{3})

= \frac{16}{9} - \frac{16}{3} = \frac{16 - 48}{9} = - \frac{32}{9}