6 x 3 8 x 2 10 x 103 2 x 4 Rewrite problem as long division 2 x 4 6 x 3 8 x 2

6 x 3 8 x 2 10 x 103 2 x 4 rewrite problem as long

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6 x 3 - 8 x 2 + 10 x + 103 2 x +4 Rewrite problem as long division 2 x +4 | 6 x 3 - 8 x 2 + 10 x + 103 Divide front terms : 6 x 3 2 x =3 x 2 3 x 2 2 x +4 | 6 x 3 - 8 x 2 + 10 x + 103 Multiply term by divisor :3 x 2 (2 x +4)=6 x 3 + 12 x 2 - 6 x 3 - 12 x 2 Change the signs and combine - 20 x 2 + 10 x Bring down the next term 3 x 2 - 10 x 2 x +4 | 6 x 3 - 8 x 2 + 10 x + 103 Repeat , divide front terms : - 20 x 2 2 x = - 10 x - 6 x 3 - 12 x 2 Multiply this term by divisor : - 20 x 2 + 10 x - 10 x (2 x +4)= - 20 x 2 - 40 x + 20 x 2 + 40 x Change the signs and combine 50 x + 103 Bring down the next term 3 x 2 - 10 x + 25 2 x +4 | 6 x 3 - 8 x 2 + 10 x + 103 Repeat , divide front terms : 50 x 2 x = 25 - 6 x 3 - 12 x 2 - 20 x 2 + 10 x + 20 x 2 + 40 x 50 x + 103 Multiply this term by divisor : 25 (2 x +4)= 50 x + 100 - 50 x - 100 Change the signs and combine 3 Remainderisputoverdivsorandadded ( duetopositive ) 3 x 2 - 10 x + 25 + 3 2 x +4 Our Solution In both of the previous examples the dividends had the exponents on our variable counting down, no exponent skipped, third power, second power, first power, zero power (remember x 0 = 1 so there is no variable on zero power). This is very important in long division, the variables must count down and no exponent can be skipped. If they don’t count down we must put them in order. If an exponent is skipped we will have to add a term to the problem, with zero for its coefficient. This is demonstrated in the following example. Page 102
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Example 6. 2 x 3 + 42 - 4 x x +3 Reorder dividend , need x 2 term , add 0 x 2 for this x +3 | 2 x 3 + 0 x 2 - 4 x + 42 Divide front terms : 2 x 3 x =2 x 2 2 x 2 x +3 | 2 x 3 +0 x 2 - 4 x + 42 Multiply this term by divisor :2 x 2 ( x +3)=2 x 3 +6 x 2 - 2 x 3 - 6 x 2 Change the signs and combine - 6 x 2 - 4 x Bring down the next term 2 x 2 - 6 x x +3 | 2 x 3 +0 x 2 - 4 x + 42 Repeat , divide front terms : - 6 x 2 x = - 6 x - 2 x 3 - 6 x 2 - 6 x 2 - 4 x Multiply this term by divisor : - 6 x ( x +3)= - 6 x 2 - 18 x +6 x 2 + 18 x Change the signs and combine 14 x + 42 Bring down the next term 2 x 2 - 6 x + 14 x +3 | 2 x 3 +0 x 2 - 4 x + 42 Repeat , divide front terms : 14 x x = 14 - 2 x 3 - 6 x 2 - 6 x 2 - 4 x +6 x 2 + 18 x 14 x + 42 Multiply this term by divisor : 14 ( x +3)= 14 x + 42 - 14 x - 42 Change the signs and combine 0 No remainder 2 x 2 - 6 x + 14 Our Solution It is important to take a moment to check each problem to verify that the exponents count down and no exponent is skipped. If so we will have to adjust the problem. Also, this final example illustrates, just as in regular long division, sometimes we have no remainder in a problem. World View Note: Paolo Ruffini was an Italian Mathematician of the early 19th century. In 1809 he was the first to describe a process called synthetic division which could also be used to divide polynomials. Page 103
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Practice - Divide Polynomials Divide. 1) 20 x 4 + x 3 +2 x 2 4 x 3 3) 20 n 4 + n 3 + 40 n 2 10 n 5) 12 x 4 + 24 x 3 +3 x 2 6 x 7) 10 n 4 + 50 n 3 +2 n 2 10 n 2 9) x 2 - 2 x - 71 x +8 11) n 2 + 13 n + 32 n +5 13) v 2 - 2 v - 89 v - 10 15) a 2 - 4 a - 38 a - 8 17) 45 p 2 + 56 p + 19 9 p +4 19) 10 x 2 - 32 x +9 10 x - 2 21) 4 r 2 - r - 1 4 r +3 23) n 2 - 4 n - 2 25) 27 b 2 + 87 b + 35 3 b +8 27) 4 x 2 - 33 x + 28 4 x - 5 29) a 3 + 15 a 2 + 49 a - 55 a +7 31) x 3 - 26 x - 41 x +4 33) 3 n 3 +9 n 2 - 64 n - 68 n +6 35) x 3 - 46 x + 22 x +7 37) 9 p 3 + 45 p 2 + 27 p - 5 9 p +9 39) r 3 - r 2 - 16 r +8 r - 4 41) 12 n 3 + 12 n 2 - 15 n - 4 2 n +3 43) 4 v 3 - 21 v 2 +6 v + 19 4 v +3 2) 5 x 4 + 45 x 3 +4 x 2 9 x 4) 3 k 3 +4 k 2 +2 k 8
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