document.write( "Question 1167825: use synthetic division to show that the given x value is a zero of the polynomial. Then find all other zeros.?
\n" ); document.write( "P(x)=x^ 3 -2x^ 2 -5x+6; x = 1 \n" ); document.write( "
Algebra.Com's Answer #792426 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Instead of doing your problem for you, I'll do another one that is\r\n" );
document.write( "EXACTLY like yours, step by step.  Use it as a guide to do yours. The\r\n" );
document.write( "problem I will do is this one:\r\n" );
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Use synthetic division to show that the given x value is a zero of the
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\n" ); document.write( "P(x)=x^3-3x^2-10x+24; x=2
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document.write( "We write down the coefficients of P(x) in a row, and the 2 from x=2\r\n" );
document.write( "to the left\r\n" );
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document.write( "  2 | 1  -3  -10  24\r\n" );
document.write( "    |               \r\n" );
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document.write( "Bring the 1 down below the bottom line:\r\n" );
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document.write( " 2 | 1   -3 -10   24\r\n" );
document.write( "   |                \r\n" );
document.write( "     1\r\n" );
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document.write( "Multiply the 1 on the bottom by the 2 at the left, getting 2.\r\n" );
document.write( "Then write it ABOVE and TO THE RIGHT of the 1 on the bottom, \r\n" );
document.write( "immediately under the -3, like this:\r\n" );
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document.write( " 2 | 1 -3 -10  24\r\n" );
document.write( "   |    2        \r\n" );
document.write( "     1\r\n" );
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document.write( "Add the -3 and the 2, getting -1, and write it below the line,\r\n" );
document.write( "like this:\r\n" );
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document.write( " 2 | 1 -3 -10  24\r\n" );
document.write( "   |    2        \r\n" );
document.write( "     1 -1\r\n" );
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document.write( "Multiply the -1 on the bottom by the 2 at the left, getting -2.\r\n" );
document.write( "Then write it ABOVE and TO THE RIGHT of the -1 on the bottom, \r\n" );
document.write( "immediately under the -10, like this: \r\n" );
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document.write( " 2 | 1 -3 -10  24\r\n" );
document.write( "   |    2  -2    \r\n" );
document.write( "     1 -1\r\n" );
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document.write( "Add the -10 and the -2 under it, getting -12, and write it below the line,\r\n" );
document.write( "like this:\r\n" );
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document.write( " 2 | 1 -3 -10  24\r\n" );
document.write( "   |    2  -2    \r\n" );
document.write( "     1 -1 -12 \r\n" );
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document.write( "Multiply the -12 on the bottom by the 2 at the left, getting -24.\r\n" );
document.write( "Then write that -24 ABOVE and TO THE RIGHT of the -12 on the bottom,\r\n" );
document.write( "under the 24, like this:\r\n" );
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document.write( " 2 | 1 -3 -10  24\r\n" );
document.write( "   |    2  -2 -24\r\n" );
document.write( "     1 -1 -12  \r\n" );
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document.write( "Add the 24 and the -24 under it, getting 0, and write the 0 below the line,\r\n" );
document.write( "under the 24, like this:\r\n" );
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document.write( " 2 | 1 -3 -10  24\r\n" );
document.write( "   |    2  -2 -24\r\n" );
document.write( "     1 -1 -12   0\r\n" );
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document.write( "That completes the synthetic division.  Now we must interpret what we have.\r\n" );
document.write( "The right-most number 0 on the bottom is the REMAINDER.\r\n" );
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document.write( "The fact that this REMAINDER number on the bottom right came out to be 0,\r\n" );
document.write( "shows that the given x value of 2 is a zero of the polynomial. \r\n" );
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document.write( "The numbers on the bottom, all except the remainder 0 on the bottom right, indicate the quotient.\r\n" );
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document.write( "We have factored this polynomial:\r\n" );
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document.write( "P(x) = x3 - 3x2 - 10x + 24\r\n" );
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document.write( "like this:\r\n" );
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document.write( "P(x) = (x - 2)(1x2 - 1x - 12)\r\n" );
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document.write( "We erase the understood 1's\r\n" );
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document.write( "P(x) = (x - 2)(x2 - x - 12)\r\n" );
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document.write( "To find all the zeros, we set the factored polynomial equal to 0\r\n" );
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document.write( "       (x - 2)(x2 - x - 12) = 0         \r\n" );
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document.write( "We use the zero-factor property, solve the x-2=0 for the zero 2\r\n" );
document.write( "that we already know we had, and factor the quadratic trinomial\r\n" );
document.write( "inside the second parentheses:\r\n" );
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document.write( "        x - 2 = 0;    x2 - x - 12 = 0\r\n" );
document.write( "            x = 2;    (x - 4)(x + 3) = 0\r\n" );
document.write( "                      x - 4 = 0;   x + 3 = 0\r\n" );
document.write( "                          x = 4        x = -3\r\n" );
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document.write( "So the other zeros besides the 2 are:   4 and -3.\r\n" );
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document.write( "Now do your problem the exact same way, step by step.\r\n" );
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document.write( "Edwin
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