document.write( "Question 255299: solve using the rational zero theorem and descarte rule:x^4-5x^3-54x^2-80x-32=0 \n" ); document.write( "

Algebra.Com's Answer #187617 by drk(1908) $\"\"$ $\"About$

You can put this solution on YOUR website!
x^4 - 5x^3 - 54x^2 - 80x - 32 = 0
\n" ); document.write( "step 1 - set up P, N, i
\n" ); document.write( "P are the number of sign changes you see when (x) is placed into the equation. We get P = 1.
\n" ); document.write( "N are the number of sign changes you see when (-x) is placed into the equation. We get N = 3.
\n" ); document.write( "Now 1 + 3 = 4.
\n" ); document.write( "P . . . 1 . . . . 1
\n" ); document.write( "N . . . 3 . . . . 1
\n" ); document.write( "i . . . .0 . . . . .2
\n" ); document.write( "Notice that i = 0 or 2. Imaginary numbers always travel in conjugate pairs.
\n" ); document.write( "Now on to rational zeros. or P/Q
\n" ); document.write( "P = +-(1, 2, 4, 8, 16, 32)
\n" ); document.write( "Q = +-(1)
\n" ); document.write( "We can find that x = -4 and x = -1 give us 2 zeros.
\n" ); document.write( "By synthetic division, using x = -4, we get
\n" ); document.write( "(x^3-9x^2-18x-8)
\n" ); document.write( "and this divided by x = -1 gets us
\n" ); document.write( "(x^2-10x-8)
\n" ); document.write( "setting this = 0 and solving gets us
\n" ); document.write( " $\"x+=+%2810+%2B-+sqrt%28+100-4%2A1%2A%28-8%29+%29%29%2F%282%29+\"$
\n" ); document.write( "or
\n" ); document.write( " $\"x+=+%2810+%2B-+sqrt%28+132+%29%29%2F%282%29+\"$
\n" ); document.write( "or simplified as
\n" ); document.write( " $\"x+=+%285+%2B-+sqrt%28+33%29%29+\"$
\n" ); document.write( "SO our four answers are:
\n" ); document.write( "x = -4, x = -1, x = 5 +-sqrt(33) \n" ); document.write( "

\n" );