Factor and solve for x

Jim Newt · #1 $Old$ 05-14-2008, 04:04 PM

x^3 + 5x^2 - x - 5 = 0

Could someone point me in the right direction as to factor and solve for x? Thanks,

Jim

Moo · #2 $Old$ 05-14-2008, 04:08 PM

Quote:

Originally Posted by Jim Newt $View Post$

x^3 + 5x^2 - x - 5 = 0

Could someone point me in the right direction as to factor and solve for x? Thanks,

Jim

Hello,

You can see that the sum of the coefficients is 0.
Hence, 1 is a root...

-> $x^3+5x^2-x-5=(x-1)P(x)$

$\begin{aligned} x^3+5x^2-x-5 &=x^3-x^2+x^2+5x^2-x-5 \\ &=x^2(x-1)+6x^2-x-5 \\ &=x^2(x-1)+6x^2-6x+6x-x-5 \\ &=x^2(x-1)+6x(x-1)+5(x-1) \\ &=(x-1)(x^2+6x+5) \end{aligned}$

Hence $x^3+5x^2-x-5=(x-1)(x+1)(x+6)$

Solve

Krizalid · #3 $Old$ 05-14-2008, 04:39 PM

$\begin{aligned} x^{3}+5x^{2}-x-5&=x^{2}\left( x+5 \right)-(x+5) \\ & =(x+5)\left( x^{2}-1 \right) \\ & =(x+5)(x+1)(x-1).\end{aligned}$

(Slight typo at the end Moo.)