if f(x) = 3x^3 + px^2 + 4x − 8 and (x − 1) is a factor of f(x). a) find the value of p. with these value of p b) solve the equation f(x) = 0. if α and β are roots of f(x), c) find α + β and αβ d) Evaluate α^2 + β^2 hence α − β


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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

	$AnOther approach$ $If f (x) is divided by x - 1 then the$ $quotient Q (x) and remainder R can$ $be determined by synthetic division :$ $(\begin{matrix} 1) 3 & p & 4 & - 8 \\ 3 & p + 3 & p + 7 \\ 3 & p + 3 & p + 7 & ∣ \overset{―}{p - 1} \end{matrix})$ $∵ x - 1 is factor of f (x)$ $∴ R = p - 1 = 0 \Rightarrow p = 1 (a)$ $Q (x) = {3 x}^{2} + (p + 3) x + (p + 7)$ $= {3 x}^{2} + (1 + 3) x + (1 + 7)$ $= {3 x}^{2} + 4 x + 8$ $f (x) = (x - 1) . Q (x) + R$ $[Relation between divedend, divisor & remainder]$ $= (x - 1) . Q (x) + 0$ $= (x - 1) ({3 x}^{2} + 4 x + 8)$ $(b) f (x) = 0 \Rightarrow x - 1 = 0 ∣ {3 x}^{2} + 4 x + 8 = 0$ $x = 1 ∣ x = \frac{- 4 \pm \sqrt{16 - 96}}{6}$ $∣ x = \frac{- 2 \pm 2 i \sqrt{5}}{3}$ $Let α & β are the roots of {3 x}^{2} + 4 x + 8 = 0$ $α + β = - \frac{4}{3} & α β = \frac{8}{3}$ $α - β = \frac{- 2 + 2 i \sqrt{5}}{3} - \frac{- 2 - 2 i \sqrt{5}}{3}$ $= \frac{4 \sqrt{5}}{3} i$ $α^{2} + β^{2} = {(α + β)}^{2} - 2 α β = {(- \frac{4}{3})}^{2} - 2 (\frac{8}{3})$ $= \frac{16}{9} - \frac{16}{3} = \frac{16 - 48}{9} = - \frac{32}{9}$
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