factorize-x-1-x-2-x-3-x-6-3x-2-

Question Number 82265 by Ali Yousafzai last updated on 19/Feb/20

f a c t o r i z e :

(x + 1) (x + 2) (x + 3) (x + 6) - 3 x^{2}

Answered by MJS last updated on 19/Feb/20

= x^{4} + 12 x^{3} + 44 x^{2} + 72 x + 36 =

[x = t - 3]

= t^{4} - 10 t^{2} + 24 t - 27 =

= (t^{2} - α t - β) (t^{2} + α t - γ) =

comparing factors leads to

{\begin{cases} α^{2} + β + γ - 10 = 0 \\ α β - α γ + 24 = 0 \\ β γ + 27 = 0 \end{cases}

\Rightarrow α = 2 \land β = - 3 \land γ = 9

= (t^{2} - 2 t + 3) (t^{2} + 2 t - 9) =

= (x^{2} + 4 x + 6) (x^{2} + 8 x + 6) =

= (x + 4 - \sqrt{10}) (x + 4 + \sqrt{10}) (x + 2 - i \sqrt{2}) (x + 2 + i \sqrt{2})

Answered by behi83417@gmail.com last updated on 19/Feb/20

(x + 1) (x + 6) = x^{2} + 7 x + 6

(x + 2) (x + 3) = x^{2} + 5 x + 6

\overset{x^{2} + 6 = t}{\Rightarrow} (t + 5 x) (t + 7 x) = t^{2} + 12 tx + {35 x}^{2}

\Rightarrow p = t^{2} + 12 tx + {32 x}^{2}

△ = {144 x}^{2} - 4 \times {32 x}^{2} = {16 x}^{2}

\Rightarrow t = \frac{- 12 x \pm 4 x}{2} = - 8 x, - 4 x

\Rightarrow p = (x^{2} + 4 x + 6) (x^{2} + 8 x + 6)

\Rightarrow p = [(x^{2} + 4 x + 4) + 2] [(x^{2} + 8 x + 16) - 10] =

= [{(x + 2)}^{2} - {2 i}^{2}] [{(x + 4)}^{2} - 10] =

= (x + 2 + i \sqrt{2}) (x + 2 - i \sqrt{2}) (x + 4 + \sqrt{10}) (x + 4 - \sqrt{10}) .