document.write( "Question 389368: How do I solve x^4+ 8x cube + 15x square 2x - 10 = 0? \n" ); document.write( "

Algebra.Com's Answer #275912 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
(Note: I am assuming that there is a minus in front of the 2x because the problem is much more difficult if there is a plus.)
\n" ); document.write( " $\"x%5E4%2B+8x%5E3+%2B+15x%5E2+-++2x+-+10+=+0\"$
\n" ); document.write( "To find solutions to this equation we will need to factor it. The only way to factor this, as far as I can see, is to use trial and error of the possible rational roots.

\n" ); document.write( "The possible rational roots of a polynomial are all the possible fractions, positive and negative, that can be formed using a factor of the constant term (at the end) over a factor of the leading coefficient (in front of the term with the highest exponent). Your constant term is -10 and you leading coefficient is 1, So the list of possible rational roots are:
\n" ); document.write( "10/1, -10/1, 5/1, -5/1, 2/1, -2/1, 1/1 and -1/1
\n" ); document.write( "which simplify to:
\n" ); document.write( "10, -10, 5, -5, 2, -2, 1 and -1

\n" ); document.write( "Synthetic division is often the method used to check these roots. This is a trial and error method but I will not waste time showing the ones that don't work. First we will try -5:
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document.write( "-5 |   1   8   15  -2   -10\r\n" );
document.write( "----      -5  -15   0    10\r\n" );
document.write( "      ---------------------\r\n" );
document.write( "       1   3    0  -2     0\r\n" );
document.write( "

\n" ); document.write( "Above we have divided $\"x%5E4%2B+8x%5E3+%2B+15x%5E2+-++2x+-+10\"$ by (x - (-5)) (or (x+5)). The zero in the lower right corner is the remainder of this division. This means (x+5) divides evenly. This means that (x+5) is a factor of $\"x%5E4%2B+8x%5E3+%2B+15x%5E2+-++2x+-+10\"$ . The rest of the bottom row tells us the other factor. The 1 3 0 -2 translates into $\"x%5E3+%2B+3x%5E2+-+2\"$ .

\n" ); document.write( "Our equation is now:
\n" ); document.write( " $\"%28x%2B5%29%28x%5E3+%2B+3x%5E2+-+2%29+=+0\"$
\n" ); document.write( "We now look for factors of $\"x%5E3+%2B+3x%5E2+-+2\"$ . Its possible rational roots are: 2, -2, 1 and -1. -1 works:
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document.write( "-1 |  1   3   0   -2\r\n" );
document.write( "----     -1  -2    2\r\n" );
document.write( "     ---------------\r\n" );
document.write( "      1   2  -2    0\r\n" );
document.write( "

\n" ); document.write( "So (x - (-1)) or (x+1) is a factor of $\"x%5E3+%2B+3x%5E2+-+2\"$ and the other factor (1 2 -2) is $\"x%5E2+%2B+2x+-+2\"$ . Our equation is now:
\n" ); document.write( " $\"%28x%2B5%29%28x%2B1%29%28x%5E2%2B2x-2%29+=+0\"$

\n" ); document.write( "The third factor is a quadratic which will not factor easily. But since it is a quadratic we can use the Quadratic Formula to find the values of x that make it zero. So we can proceed to the next step.

\n" ); document.write( "The Zero Product Property tells us that this (or any) product can be zero only if one (or more) of the factors is zero. So:
\n" ); document.write( "x+5 = 0 or x+1 = 0 or $\"x%5E2%2B3x-2+=+0\"$
\n" ); document.write( "The first two are simple to solve. We get x = -5 and x = -1 for solutions. The third equation requires use of the Quadratic Formula:
\n" ); document.write( " $\"x+=+%28-%282%29+%2B-+sqrt%28%282%29%5E2+-+4%281%29%28-2%29%29%29%2F2%281%29\"$
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( " $\"x+=+%28-%282%29+%2B-+sqrt%284+-+4%281%29%28-2%29%29%29%2F2%281%29\"$
\n" ); document.write( " $\"x+=+%28-%282%29+%2B-+sqrt%284+%2B+8%29%29%2F2%281%29\"$
\n" ); document.write( " $\"x+=+%28-%282%29+%2B-+sqrt%2812%29%29%2F2%281%29\"$
\n" ); document.write( " $\"x+=+%28-2+%2B-+sqrt%2812%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%28-2+%2B-+sqrt%284%2A3%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%28-2+%2B-+sqrt%284%29%2Asqrt%283%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%28-2+%2B-+2%2Asqrt%283%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%282%28-1+%2B-+sqrt%283%29%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%28cross%282%29%28-1+%2B-+sqrt%283%29%29%29%2Fcross%282%29\"$
\n" ); document.write( " $\"x+=+-1+%2B-+sqrt%283%29\"$
\n" ); document.write( "In long form this is:
\n" ); document.write( " $\"x+=+-1+%2B+sqrt%283%29\"$ or $\"x+=+-1+-+sqrt%283%29\"$

\n" ); document.write( "So the solutions to your equation are:
\n" ); document.write( "x = -5 or x = -1 or $\"x+=+-1+%2B+sqrt%283%29\"$ or $\"x+=+-1+-+sqrt%283%29\"$ \n" ); document.write( "

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