document.write( "Question 1037052: Use synthetic substitution to find f(2) and f(3).
\n" ); document.write( "f(x) = 3x⁴ - 12x³ - 12x² + 30 x
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #651743 by Edwin McCravy(20060) $\"\"$ $\"About$
You can put this solution on YOUR website!
\r\n" );
document.write( "3x⁴ - 12x³ - 12x² + 30x\r\n" );
document.write( "\r\n" );
document.write( "That's missing a constant term so we put + 0\r\n" );
document.write( "on the end:\r\n" );
document.write( "\r\n" );
document.write( "3x⁴ - 12x³ - 12x² + 30x + 0\r\n" );
document.write( "\r\n" );
document.write( "We erase all the letters and exponents\r\n" );
document.write( "\r\n" );
document.write( "3 -12 -12 30  0\r\n" );
document.write( "\r\n" );
document.write( "To find f(2) by synthetic substitution:\r\n" );
document.write( "Put 2 on the left and draw a line:\r\n" );
document.write( "\r\n" );
document.write( "2 | 3  -12  -12  30  0\r\n" );
document.write( "  |___________________     \r\n" );
document.write( "    \r\n" );
document.write( "Bring down the 3:\r\n" );
document.write( "\r\n" );
document.write( "2 | 3  -12  -12  30  0\r\n" );
document.write( "  |___________________\r\n" );
document.write( "    3\r\n" );
document.write( "\r\n" );
document.write( "Multiply the 3 on the bottom by the 2 at the left.\r\n" );
document.write( "Get 6.  Put the 6 above and to the right of the 3,\r\n" );
document.write( "right under the first -12\r\n" );
document.write( "\r\n" );
document.write( "2 | 3  -12  -12  30  0\r\n" );
document.write( "  |      6             \r\n" );
document.write( "    3\r\n" );
document.write( "\r\n" );
document.write( "Now combine the -12 and the 6, get -6, and write\r\n" );
document.write( "the -6 down below the bottom line, right of the 3\r\n" );
document.write( "and underneath the 6.\r\n" );
document.write( "\r\n" );
document.write( "2 | 3  -12  -12  30  0\r\n" );
document.write( "  |      6             \r\n" );
document.write( "    3   -6\r\n" );
document.write( "\r\n" );
document.write( "Multiply the -6 on the bottom by the 2 at the left.\r\n" );
document.write( "Get -12.  Put the -12 above and to the right of the -6,\r\n" );
document.write( "right under the second -12 on the top row:\r\n" );
document.write( "\r\n" );
document.write( "2 | 3  -12  -12  30  0\r\n" );
document.write( "  |      6  -12        \r\n" );
document.write( "    3   -6\r\n" );
document.write( "\r\n" );
document.write( "Now combine the two -12's, getting -24, and write\r\n" );
document.write( "the -24 down below the bottom line, right of the -6\r\n" );
document.write( "and underneath the -12.\r\n" );
document.write( "\r\n" );
document.write( "2 | 3  -12  -12  30  0\r\n" );
document.write( "  |      6  -12        \r\n" );
document.write( "    3   -6  -24\r\n" );
document.write( "\r\n" );
document.write( "Multiply the -24 on the bottom by the 2 at the left.\r\n" );
document.write( "Get -48.  Put the -48 above and to the right of the -24,\r\n" );
document.write( "right under 30 on the top row:\r\n" );
document.write( "\r\n" );
document.write( "2 | 3  -12  -12  30  0\r\n" );
document.write( "  |      6  -12 -48    \r\n" );
document.write( "    3   -6  -24 \r\n" );
document.write( "\r\n" );
document.write( "Now combine the 30 and the -48, getting -18, and write\r\n" );
document.write( "the -18 down below the bottom line, right of the -24\r\n" );
document.write( "and underneath the -48.\r\n" );
document.write( "\r\n" );
document.write( "2 | 3  -12  -12  30  0\r\n" );
document.write( "  |      6  -12 -48    \r\n" );
document.write( "    3   -6  -24 -18\r\n" );
document.write( "\r\n" );
document.write( "Multiply the -18 on the bottom by the 2 at the left.\r\n" );
document.write( "Get -36.  Put the -36 above and to the right of the -18,\r\n" );
document.write( "right under 0 on the top row:\r\n" );
document.write( "\r\n" );
document.write( "2 | 3  -12  -12  30   0\r\n" );
document.write( "  |      6  -12 -48 -36 \r\n" );
document.write( "    3   -6  -24 -18\r\n" );
document.write( " \r\n" );
document.write( "Finally combine the 0 and the -36, getting -36, and write\r\n" );
document.write( "the -36 down below the bottom line, right of the -18\r\n" );
document.write( "and underneath the -36, in the bottom right corner:\r\n" );
document.write( "\r\n" );
document.write( "2 | 3  -12  -12  30   0\r\n" );
document.write( "  |      6  -12 -48 -36 \r\n" );
document.write( "    3   -6  -24 -18 -36\r\n" );
document.write( "\r\n" );
document.write( "Now it's finished, and f(2) = -36 (the number in the bottom\r\n" );
document.write( "right corner.\r\n" );
document.write( "\r\n" );
document.write( "Check synthetic substitution by direct substitution: \r\n" );
document.write( " \r\n" );
document.write( "3x⁴ - 12x³ - 12x² + 30x\r\n" );
document.write( "\r\n" );
document.write( "3(2)⁴ - 12(2)³ - 12(2)² + 30(2)\r\n" );
document.write( "\r\n" );
document.write( "3(16) - 12(8) - 12(4) + 60\r\n" );
document.write( "\r\n" );
document.write( "48 - 96 - 48 + 60\r\n" );
document.write( "\r\n" );
document.write( "-48 - 48 + 60\r\n" );
document.write( "\r\n" );
document.write( "-96 + 60\r\n" );
document.write( "\r\n" );
document.write( "-36.\r\n" );
document.write( "\r\n" );
document.write( "So it's correct since direct substitution\r\n" );
document.write( "gives the same answer.\r\n" );
document.write( "\r\n" );
document.write( "Now you find f(3) the exact same way.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( " \n" ); document.write( "
\n" );