factor-x-3-x-2-x-1-3-and-x-3-x-2-x-1-3-and-x-3-x-2-x-1-3-

Question Number 50925 by kaivan.ahmadi last updated on 22/Dec/18

factor x^{3} + x^{2} + x - \frac{1}{3} and

x^{3} + x^{2} - x + \frac{1}{3} and

x^{3} + x^{2} - x - \frac{1}{3}

Commented by Abdo msup. last updated on 23/Dec/18

l e t p (x) = x^{3} + x^{2} + x - \frac{1}{3} l e t f i n d t h e r o o t s c h a n g e m e n t x = t - \frac{1}{3}

g i v e p (x) = {(t - \frac{1}{3})}^{3} + {(t - \frac{1}{3})}^{2} + t - \frac{2}{3}

= t^{3} - 3 t^{2} . \frac{1}{3} + 3 t \frac{1}{9} - \frac{1}{27} + t^{2} - \frac{2}{3} t + \frac{1}{9} + t - \frac{2}{3}

= t^{3} - t^{2} + \frac{t}{3} + t^{2} + \frac{1}{3} t + \frac{- 1 + 3 - 18}{27}

= t^{3} + \frac{2}{3} t - \frac{16}{27} = g (t) l e t s o l v e g (t) = 0 w i t h t = u + v

w e g e t {(u + v)}^{3} + \frac{2}{3} (u + v) - \frac{16}{27} = 0 \Rightarrow

u^{3} + v^{3} + 3 u v (u + v) + \frac{2}{3} (u + v) - \frac{16}{27} = 0 \Rightarrow

u^{3} + v^{3} - \frac{16}{27} = 0 a n d 3 u v + \frac{2}{3} = 0 \Rightarrow

u^{3} + v^{3} = \frac{16}{27} a n d u v = - \frac{2}{9} \Rightarrow

u^{3} + v^{3} = \frac{16}{27} a n d u^{3} v^{3} = \frac{- 8}{729} s o u^{3} a n d v^{3} a r e s o l u t i o n

o f t h e e q u a t i o n z^{2} - \frac{16}{27} z - \frac{8}{729} = 0 \dots

Δ = {(- \frac{16}{27})}^{2} + \frac{32}{729} > 0 \Rightarrow z_{1} = \frac{\frac{16}{27} + \sqrt{\frac{16^{2}}{27^{2}} + \frac{32}{729}}}{2}

z_{2} = \dots b e c o n t i n u e d \dots