# Solve x^4 - x^3 + 8x - 8 = 0, help pls

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• Mar 16th 2008, 06:49 AM
zxe
Solve x^4 - x^3 + 8x - 8 = 0, help pls
How to solve or factorize this equation?
\$\displaystyle x^4 - x^3 + 8x - 8 = 0\$

help please
• Mar 16th 2008, 06:55 AM
Moo
Hello,

You can observe that the sum of the coefficients is 0. So 1 is solution. This means that you can factorise by (x-1).

So \$\displaystyle x^4-x^3+8x-8 = (x-1)Q(x)\$

Where Q(x) is a polynom of degree 3 -> Q(x)=ax^3 + bx² + cx + d, with a,b,c,d to determine by developping (x-1)Q(x).
• Mar 16th 2008, 06:55 AM
wingless
\$\displaystyle x^4 - x^3 + 8x - 8 = 0\$

\$\displaystyle x^3(x-1) + 8(x-1) = 0\$

\$\displaystyle (x-1)(x^3+8) = 0\$

---> \$\displaystyle x-1 = 0\$
---> \$\displaystyle x^3+8 = 0\$

\$\displaystyle x=\{-2,1\}\$
• Mar 16th 2008, 08:09 AM
Jhevon
Quote:

Originally Posted by wingless
\$\displaystyle x^4 - x^3 + 8x - 8 = 0\$

\$\displaystyle x^3(x-1) + 8(x-1) = 0\$

\$\displaystyle (x-1)(x^3+8) = 0\$

---> \$\displaystyle x-1 = 0\$
---> \$\displaystyle x^3+8 = 0\$

\$\displaystyle x=\{-2,1\}\$

wingless is right. i'd just like to point out that we can solve \$\displaystyle x^2 + 8 = 0\$ using the sum of two cubes formula (if you don't see the solution immediately). so, \$\displaystyle x^3 + 8 = (x + 2)(x^2 - 2x + 2^2) = 0\$, and now equate each term in the product to zero and solve.
• Mar 16th 2008, 12:28 PM
topsquark
Quote:

Originally Posted by wingless
\$\displaystyle x^4 - x^3 + 8x - 8 = 0\$

\$\displaystyle x^3(x-1) + 8(x-1) = 0\$

\$\displaystyle (x-1)(x^3+8) = 0\$

---> \$\displaystyle x-1 = 0\$
---> \$\displaystyle x^3+8 = 0\$

\$\displaystyle x=\{-2,1\}\$

You missed out on two complex solutions.
\$\displaystyle x^4 - x^3 + 8x - 8 = 0\$

\$\displaystyle (x - 1)(x^3 + 8) = 0\$

\$\displaystyle (x - 1)(x + 2)(x^2 - 2x + 4) = 0\$

Now let each factor be zero and solve.

-Dan