document.write( "Question 1183635: Given that P(x) = x^4 + ax^3 - x^2 + bx - 12 has factors x-2 and x+1,solve the equation P(x) = 0. \n" ); document.write( "

Algebra.Com's Answer #814035 by greenestamps(13203) $\"\"$ $\"About$

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\n" ); document.write( "Since x-2 is a factor of P(x), P(2) is 0:
\n" ); document.write( " $\"16%2B8a-4%2B2b-12=0\"$
\n" ); document.write( " $\"8a%2B2b=0\"$
\n" ); document.write( " $\"4a%2Bb=0\"$
\n" ); document.write( " $\"b=-4a\"$ [1]

\n" ); document.write( "Since x+1 is a factor of P(x), P(-1)is 0:
\n" ); document.write( " $\"1-a-1-b-12=0\"$
\n" ); document.write( " $\"a%2Bb=-12\"$ [2]

\n" ); document.write( "Solving [1] and [2] gives a=4, from which b=-16

\n" ); document.write( "So the polynomial is

\n" ); document.write( " $\"P%28x%29=x%5E4%2B4x%5E3-x%5E2-16x-12\"$

\n" ); document.write( "Use synthetic division to factor out the two known linear factors:

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document.write( "  2 | 1  4  -1 -16 -12\r\n" );
document.write( "    |    2  12  22  12\r\n" );
document.write( "    ------------------\r\n" );
document.write( " -1 | 1  6  11   6   0\r\n" );
document.write( "    |   -1  -5  -6\r\n" );
document.write( "    --------------\r\n" );
document.write( "      1  5   6   0

\n" ); document.write( "The reduced polynomial is

\n" ); document.write( " $\"x%5E2%2B5x%2B6+=+%28x%2B2%29%28x%2B3%29\"$

\n" ); document.write( "So the other two roots are -2 and -3.

\n" ); document.write( "ANSWER: P(x)=0 --> x = -3, -2, -1, and 2

\n" ); document.write( "A graph confirms those roots:

\n" ); document.write( " $\"graph%28400%2C400%2C-5%2C5%2C-20%2C20%2Cx%5E4%2B4x%5E3-x%5E2-16x-12%29\"$

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