Example 1 9 x 5 6 x 4 18 x 3 24 x 2 3 x 2 Divide each term in the numerator by

Example 1 9 x 5 6 x 4 18 x 3 24 x 2 3 x 2 divide each

This preview shows page 99 - 103 out of 200 pages.

Example 1. 9 x 5 +6 x 4 - 18 x 3 - 24 x 2 3 x 2 Divide each term in the numerator by 3 x 2 9 x 5 3 x 2 + 6 x 4 3 x 2 - 18 x 3 3 x 2 - 24 x 2 3 x 2 Reduce each fraction , subtracting exponents 3 x 3 +2 x 2 - 6 x - 8 Our Solution Example 2. 8 x 3 +4 x 2 - 2 x +6 4 x 2 Divide each term in the numerator by 4 x 2 8 x 3 4 x 2 + 4 x 2 4 x 2 - 2 x 4 x 2 + 6 4 x 2 Reduce each fraction , subtracting exponents Remember negative exponents are moved to denominator 2 x +1 - 1 2 x + 3 2 x 2 Our Solution The previous example illustrates that sometimes we will have fractions in our solution, as long as they are reduced this will be correct for our solution. Also interesting in this problem is the second term 4 x 2 4 x 2 divided out completely. Remember that this means the reduced answer is 1 not 0. Long division is required when we divide by more than just a monomial. Long division with polynomials works very similarly to long division with whole numbers. An example is given to review the (general) steps that are used with whole numbers that we will also use with polynomials Page 99
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Example 3. 4 | 631 Divide front numbers : 6 4 =1 ... 1 4 | 631 Multiply this number by divisor :1 · 4=4 - 4 Change the sign of this number ( make it subtract ) and combine 23 Bring down next number 1 5 Repeat , divide front numbers : 23 4 =5 ... 4 | 631 - 4 23 Multiply this number by divisor :5 · 4= 20 - 20 Change the sign of this number ( make it subtract ) and combine 31 Bring down next number 15 7 Repeat , divide front numbers : 31 4 =7 ... 4 | 631 - 4 23 - 20 31 Multiply this number by divisor :7 · 4= 28 - 28 Change the sign of this number ( make it subtract ) and combine 3 We will write our remainder as a fraction , over the divisor , added to the end 157 3 4 Our Solution This same process will be used to divide polynomials. The only difference is we will replace the word “number” with the word “term”. Dividing Polynomials 1. Divide front terms 2. Multiply this term by the divisor 3. Change the sign of the terms and combine Page 100
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4. Bring down the next term 5. Repeat Step number 3 tends to be the one that students skip, not changing the signs of the terms would be equivalent to adding instead of subtracting on long division with whole numbers. Be sure not to miss this step! This process is illustrated in the following two examples. Example 4. 3 x 3 - 5 x 2 - 32 x +7 x - 4 Rewrite problem as long division x - 4 | 3 x 3 - 5 x 2 - 32 x +7 Divide front terms : 3 x 3 x =3 x 2 3 x 2 x - 4 | 3 x 3 - 5 x 2 - 32 x +7 Multiply this term by divisor :3 x 2 ( x - 4)=3 x 3 - 12 x 2 - 3 x 3 + 12 x 2 Change the signs and combine 7 x 2 - 32 x Bring down the next term 3 x 2 + 7 x Repeat , divide front terms : 7 x 2 x =7 x x - 4 | 3 x 3 - 5 x 2 - 32 x +7 - 3 x 3 + 12 x 2 7 x 2 - 32 x Multiply this term by divisor :7 x ( x - 4)=7 x 2 - 28 x - 7 x 2 + 28 x Change the signs and combine - 4 x +7 Bring down the next term 3 x 2 +7 x - 4 Repeat , divide front terms : - 4 x x = - 4 x - 4 | 3 x 3 - 5 x 2 - 32 x +7 - 3 x 3 + 12 x 2 7 x 2 - 32 x - 7 x 2 + 28 x - 4 x +7 Multiply this term by divisor : - 4( x - 4)= - 4 x + 16 +4 x - 16 Change the signs and combine - 9 Remainderputoverdivisorandsubtracted ( duetonegative ) 3 x 2 +7 x - 4 - 9 x - 4 Our Solution Page 101
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Example 5.
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