factorize-2x-3-1-

Question Number 67136 by ,jamiebots last updated on 23/Aug/19

f a c t o r i z e 2 x^{3} - 1

Answered by Cmr 237 last updated on 23/Aug/19

{2 x}^{3} - 1 = 2 (x^{3} - \frac{1}{2}) = p (x)

= 2 (x - \sqrt[3]{\frac{1}{2}}) (x^{2} + \sqrt[3]{\frac{1}{2}} x + {(\sqrt[3]{\frac{1}{2}})}^{2})

or (x^{2} + \sqrt[3]{\frac{1}{2}} x + {(\sqrt[3]{\frac{1}{2}})}^{2}) = 0 \Rightarrow

Δ = - 3 {(\sqrt[3]{\frac{1}{2}})}^{2} = {(\sqrt{3} i \sqrt[3]{\frac{1}{2}})}^{2}

x_{1} = - \frac{\sqrt[3]{\frac{1}{2}} + \sqrt{3} i \sqrt[3]{\frac{1}{2}}}{2}

x_{2} = - \frac{\sqrt[3]{\frac{1}{2}} - \sqrt{3} i \sqrt[3]{\frac{1}{2}}}{2}

ainsi,

p (x) = 2 (x - x_{0}) (x - x_{1}) (x - x_{2})

avec; x_{0} = \sqrt[3]{\frac{1}{2}} .

Answered by Rasheed.Sindhi last updated on 23/Aug/19

2 x^{3} - 1 = {(^{3} \sqrt{2} x)}^{3} - {(1)}^{3}

= (^{3} \sqrt{2} x - 1) ({(^{3} \sqrt{2} x)}^{2} + (^{3} \sqrt{2} x) (1) + {(1)}^{2})

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

F o r m u l a :

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})