document.write( "Question 146421: Can someone please help me with a couple of problems on polynomials finding the quotient and the remainder?\r
\n" ); document.write( "\n" ); document.write( "1) p(x)= 2x3-3x2+x-1
\n" ); document.write( " d(x) = x-3\r
\n" ); document.write( "\n" ); document.write( "2) P(x) = x3 - 9x2 + 15x + 25
\n" ); document.write( " d(x) = x-5\r
\n" ); document.write( "\n" ); document.write( "3) P(x) = x4 + 6x3
\n" ); document.write( " d(x) = x-1 \n" ); document.write( "
Algebra.Com's Answer #106907 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
Can someone please help me with a couple of problems on polynomials finding the quotient and the remainder?
\n" ); document.write( "1) \"p%28x%29=+2x3-3x2%2Bx-1\"
\n" ); document.write( "\"d%28x%29+=+x-3\"
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\r\n" );
document.write( "By long division you would start with\r\n" );
document.write( "     ___________________\r\n" );
document.write( "x - 3) 2x3 - 3x2 + x - 1 \r\n" );
document.write( "\r\n" );
document.write( "But with synthetic division\r\n" );
document.write( "\r\n" );
document.write( "Change the sign of the -3 to +3\r\n" );
document.write( "and write this:\r\n" );
document.write( "\r\n" );
document.write( "   3 |\r\n" );
document.write( "     |__________ \r\n" );
document.write( "\r\n" );
document.write( "Then list the coefficients of the terms:\r\n" );
document.write( "\r\n" );
document.write( "   3 |2 -3  1 -1\r\n" );
document.write( "     |__________   \r\n" );
document.write( "     \r\n" );
document.write( "\r\n" );
document.write( "Start by bringing down the 2\r\n" );
document.write( "\r\n" );
document.write( "   3 |2 -3  1 -1\r\n" );
document.write( "     |__________  \r\n" );
document.write( "      2 \r\n" );
document.write( "\r\n" );
document.write( "Multiply the 2 at the bottom by the 3 at the left, \r\n" );
document.write( "getting 6. Write that 6 above and to the right of \r\n" );
document.write( "the 2, under the next term -3:\r\n" );
document.write( "\r\n" );
document.write( "   3 |2 -3  1 -1\r\n" );
document.write( "     |   6        \r\n" );
document.write( "      2  \r\n" );
document.write( "\r\n" );
document.write( "Combine the -3 and the 6, getting 3, and write this\r\n" );
document.write( "below the line under the 6:\r\n" );
document.write( "\r\n" );
document.write( "   3 |2 -3  1 -1\r\n" );
document.write( "     |   6        \r\n" );
document.write( "      2  3  \r\n" );
document.write( "\r\n" );
document.write( "Multiply that 3 on the bottom by the 3 at the left,\r\n" );
document.write( "getting 9, and write this 9 above and to the right of the\r\n" );
document.write( "3, above the line below the 1: \r\n" );
document.write( "\r\n" );
document.write( "   3 |2 -3  1 -1\r\n" );
document.write( "     |   6  9    \r\n" );
document.write( "      2  3  \r\n" );
document.write( "\r\n" );
document.write( "Combine the 1 and the 9, getting 10, and write this\r\n" );
document.write( "below the line under the 9:\r\n" );
document.write( "\r\n" );
document.write( "   3 |2 -3  1 -1\r\n" );
document.write( "     |   6  9   \r\n" );
document.write( "      2  3 10 \r\n" );
document.write( "\r\n" );
document.write( "Multiply 10 on the bottom by the 3 at the left, getting 30,\r\n" );
document.write( "and write this 30 above and to the right of the 10, above \r\n" );
document.write( "the line below the -1:\r\n" );
document.write( "\r\n" );
document.write( "   3 |2 -3  1 -1\r\n" );
document.write( "     |   6  9 30\r\n" );
document.write( "      2  3 10 \r\n" );
document.write( "\r\n" );
document.write( "Combine the -1 and the 30, getting 29, and write this\r\n" );
document.write( "29 below the line under the 30:\r\n" );
document.write( "\r\n" );
document.write( "   3 |2 -4  1 -1\r\n" );
document.write( "     |   6  9 30\r\n" );
document.write( "      2  3 10 29\r\n" );
document.write( "\r\n" );
document.write( "Now we must interpret that row of numbers 2  3 10 29\r\n" );
document.write( "across the bottom of the synthetic division.\r\n" );
document.write( "\r\n" );
document.write( "The largest power of x in the original polynomial is 3.\r\n" );
document.write( "So the 2 is multiplied by a power of x which is 1 lower \r\n" );
document.write( "than the degree of the original polynomial. One lower \r\n" );
document.write( "than 3 is 2, so we write x2 after the 2. Then we \r\n" );
document.write( "write x (first power) after the 3. Then 10 is the \r\n" );
document.write( "constant term.  So the quotient only is\r\n" );
document.write( "\r\n" );
document.write( "     2x2 + 3x + 10\r\n" );
document.write( "\r\n" );
document.write( "and the last number on the bottom right is the remainder.  So\r\n" );
document.write( "we put the remainder 29 over the divisor x - 3  and we have\r\n" );
document.write( "as the final answer:\r\n" );
document.write( "\r\n" );
document.write( "    \"2x%5E2%2B3x%2B10%2B29%2F%28x-3%29\"\r\n" );
document.write( "\r\n" );
document.write( "==========================================================\r\n" );
document.write( "\r\n" );
document.write( "2) \"P%28x%29+=+x%5E3+-+9x%5E2+%2B+15x+%2B+25\"\r\n" );
document.write( "d(x) = \"x-5\" \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "By long division you would start with\r\n" );
document.write( "     _____________________\r\n" );
document.write( "x - 5) x3 - 9x2 + 15x + 25 \r\n" );
document.write( "\r\n" );
document.write( "But with synthetic division\r\n" );
document.write( "\r\n" );
document.write( "Change the sign of the -5 to +5\r\n" );
document.write( "and write this:\r\n" );
document.write( "\r\n" );
document.write( "   5 |\r\n" );
document.write( "     |__________ \r\n" );
document.write( "\r\n" );
document.write( "Then list the coefficients of the terms:\r\n" );
document.write( "\r\n" );
document.write( "   5 |1 -9 15 25\r\n" );
document.write( "     |__________   \r\n" );
document.write( "     \r\n" );
document.write( "\r\n" );
document.write( "Start by bringing down the 1\r\n" );
document.write( "\r\n" );
document.write( "   5 |1 -9 15 25\r\n" );
document.write( "     |__________  \r\n" );
document.write( "      1 \r\n" );
document.write( "\r\n" );
document.write( "Multiply the 1 at the bottom by the 5 at the left, \r\n" );
document.write( "getting 5. Write that 5 above and to the right of \r\n" );
document.write( "the 1, under the next term -9:\r\n" );
document.write( "\r\n" );
document.write( "   5 |1 -9 15 25\r\n" );
document.write( "     |   5        \r\n" );
document.write( "      1  \r\n" );
document.write( "\r\n" );
document.write( "Combine the -9 and the 5, getting -4, and write this\r\n" );
document.write( "below the line under the 5:\r\n" );
document.write( "\r\n" );
document.write( "   5 |1 -9 15 25\r\n" );
document.write( "     |   5        \r\n" );
document.write( "      1 -4  \r\n" );
document.write( "\r\n" );
document.write( "Multiply that -4 on the bottom by the 5 at the left,\r\n" );
document.write( "getting -20, and write this -20 above and to the right of the\r\n" );
document.write( "-4, above the line below the 15: \r\n" );
document.write( "\r\n" );
document.write( "   5 |1 -9 15 25\r\n" );
document.write( "     |   5-20    \r\n" );
document.write( "      1 -4  \r\n" );
document.write( "\r\n" );
document.write( "Combine the 15 and the -20, getting -5, and write this\r\n" );
document.write( "below the line under the -20:\r\n" );
document.write( "\r\n" );
document.write( "   5 |1 -9 15 25\r\n" );
document.write( "     |   5-20   \r\n" );
document.write( "      1 -4 -5 \r\n" );
document.write( "\r\n" );
document.write( "Multiply -5 on the bottom by the 5 at the left, getting -25,\r\n" );
document.write( "and write this -25 above and to the right of the -5, above \r\n" );
document.write( "the line below the -20:\r\n" );
document.write( "\r\n" );
document.write( "   5 |1 -9 15 25\r\n" );
document.write( "     |   5-20-25\r\n" );
document.write( "      1 -4 1-5  \r\n" );
document.write( "\r\n" );
document.write( "Combine the 25 and the -25, getting 0, and write this\r\n" );
document.write( "0 below the line under the -25:\r\n" );
document.write( "\r\n" );
document.write( "   5 |1 -9 15 25\r\n" );
document.write( "     |   5-20-25\r\n" );
document.write( "      1 -4 -5  0\r\n" );
document.write( "\r\n" );
document.write( "Now we must interpret that row of numbers 1 -4 -5  0\r\n" );
document.write( "across the bottom of the synthetic division.\r\n" );
document.write( "\r\n" );
document.write( "The largest power of x in the original polynomial is 3.\r\n" );
document.write( "So the 1 is multiplied by a power of x which is 1 lower \r\n" );
document.write( "than the degree of the original polynomial. One lower \r\n" );
document.write( "than 3 is 2, so we write x2 after the 1. Then we \r\n" );
document.write( "write x (first power) after the -4. Then -5 is the \r\n" );
document.write( "constant term.  So the quotient is\r\n" );
document.write( "\r\n" );
document.write( "      x2 - 4x - 5\r\n" );
document.write( "\r\n" );
document.write( "and the last number on the bottom right is the remainder. \r\n" );
document.write( "Since that number is 0, that is the final answer.\r\n" );
document.write( "\r\n" );
document.write( "-----------------------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "3) P(x) = x4 + 6x3\r\n" );
document.write( "d(x) = x-1\r\n" );
document.write( "\r\n" );
document.write( "Nere we must use placeholders for x2, x1, and the constant\r\n" );
document.write( "term\r\n" );
document.write( "\r\n" );
document.write( "By long division you would start with\r\n" );
document.write( "     _______________________\r\n" );
document.write( "x - 1) x4 + 6x3 + 0x + 0 + 0 \r\n" );
document.write( "\r\n" );
document.write( "But with synthetic division\r\n" );
document.write( "\r\n" );
document.write( "Change the sign of the -1 to +1\r\n" );
document.write( "and write this:\r\n" );
document.write( "\r\n" );
document.write( "   1 | 1   6   0   0   0\r\n" );
document.write( "     |__________________\r\n" );
document.write( "       \r\n" );
document.write( "Start by bringing down the 1\r\n" );
document.write( "\r\n" );
document.write( "   1 | 1   6   0   0   0\r\n" );
document.write( "     |__________________\r\n" );
document.write( "       1\r\n" );
document.write( " \r\n" );
document.write( "Multiply the 1 at the bottom by the 1 at the left, \r\n" );
document.write( "getting 1. Write that 1 above and to the right of \r\n" );
document.write( "the 1 at the bottom, under the next term 6:\r\n" );
document.write( "\r\n" );
document.write( "   1 | 1   6   0   0   0\r\n" );
document.write( "     |_____1____________\r\n" );
document.write( "       1\r\n" );
document.write( "\r\n" );
document.write( "Combine the 6 and the 1, getting 7, and write this\r\n" );
document.write( "7 below the line under the 1:\r\n" );
document.write( "\r\n" );
document.write( "   1 | 1   6   0   0   0\r\n" );
document.write( "     |_____1____________\r\n" );
document.write( "       1   7\r\n" );
document.write( "\r\n" );
document.write( "Multiply the 7 at the bottom by the 1 at the left, \r\n" );
document.write( "getting 7. Write that 7 above and to the right of \r\n" );
document.write( "the 7 at the bottom, under the next term 0:\r\n" );
document.write( "\r\n" );
document.write( "   1 | 1   6   0   0   0\r\n" );
document.write( "     |_____1___7________\r\n" );
document.write( "       1   7\r\n" );
document.write( "\r\n" );
document.write( "Combine the 0 and the 7, getting 7, and write this\r\n" );
document.write( "7 below the line under the 7:\r\n" );
document.write( "\r\n" );
document.write( "   1 | 1   6   0   0   0\r\n" );
document.write( "     |_____1___7________\r\n" );
document.write( "       1   7   7\r\n" );
document.write( "\r\n" );
document.write( "Multiply the second 7 at the bottom by the 1 at the \r\n" );
document.write( "left, getting 7. Write that 7 above and to the right of \r\n" );
document.write( "the second 7 at the bottom, under the next term 0:\r\n" );
document.write( "\r\n" );
document.write( "   1 | 1   6   0   0   0\r\n" );
document.write( "     |_____1___7___7____\r\n" );
document.write( "       1   7   7\r\n" );
document.write( "\r\n" );
document.write( "Combine the 0 and the 7, getting 7, and write this\r\n" );
document.write( "7 below the line under the 7:\r\n" );
document.write( "\r\n" );
document.write( "   1 | 1   6   0   0   0\r\n" );
document.write( "     |_____1___7___7____\r\n" );
document.write( "       1   7   7   7\r\n" );
document.write( "\r\n" );
document.write( "Multiply the third 7 at the bottom by the 1 at the \r\n" );
document.write( "left, getting 7. Write that 7 above and to the right of \r\n" );
document.write( "the third 7 at the bottom, under the last term 0:\r\n" );
document.write( "\r\n" );
document.write( "   1 | 1   6   0   0   0\r\n" );
document.write( "     |_____1___7___7___7\r\n" );
document.write( "       1   7   7   7\r\n" );
document.write( "\r\n" );
document.write( "Combine the 0 and the 7, getting 7, and write this\r\n" );
document.write( "7 below the line under the 7:\r\n" );
document.write( "\r\n" );
document.write( "   1 | 1   6   0   0   0\r\n" );
document.write( "     |_____1___7___7___7\r\n" );
document.write( "       1   7   7   7   7\r\n" );
document.write( "\r\n" );
document.write( "Now we must interpret that row of numbers 1   7   7   7   7\r\n" );
document.write( "across the bottom of the synthetic division.\r\n" );
document.write( "\r\n" );
document.write( "The largest power of x in the original polynomial is 4.\r\n" );
document.write( "So the 1 is multiplied by a power of x which is 1 lower \r\n" );
document.write( "than the degree of the original polynomial. One lower \r\n" );
document.write( "than 4 is 3, so we write x3 after the 1. Then we \r\n" );
document.write( "write x2 after the first 7. Then x after the second 7.\r\n" );
document.write( "The third 7 on the bottom is the constant term.  The\r\n" );
document.write( "quotient only is\r\n" );
document.write( "\r\n" );
document.write( "     x³ + 7x² + 7x + 7\r\n" );
document.write( "\r\n" );
document.write( "and the last number on the bottom right is the remainder.  So\r\n" );
document.write( "we put the remainder 7 over the divisor x - 1  and we have\r\n" );
document.write( "as the final answer:\r\n" );
document.write( "\r\n" );
document.write( "    \"x%5E3%2B7x%5E2%2B7x%2B7%2B7%2F%28x-1%29\"\r\n" );
document.write( "\r\n" );
document.write( "Edwin
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