Synthetic division

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Synthetic division

  1. 1. Let’s look at how to do this using the example: ( 5x #1 4 − 4 x + x + 6 ) ÷ ( x − 3) 2 In order to use synthetic division these two things must happen: #2 The divisor must There must be a coefficient for have a leading every possible coefficient of 1. power of the variable.
  2. 2. Step #1: Write the terms of the polynomial so the degrees are in descending order. 5x + 0x − 4x + x + 6 4 3 2 Since the numerator does not contain all the powers of x, you must include a 0 for the x 3 .
  3. 3. Step #2: Write the constant a of the divisor x- a to the left and write down the coefficients. Since the divisor = x − 3, then a = 3 5x 4 0x 3 −4 x 2 + x +6 ↓ 3 ↓ ↓ ↓ ↓ 5 0 −4 1 6
  4. 4. Step #3: Bring down the first coefficient, 5. 3 5 0 −4 1 6 ↓ 5 Step #4: Multiply the first coefficient by r (3*5). 3 5 0 ↓ 15 5 −4 1 6
  5. 5. Step #5: After multiplying in the diagonals, add the column. Add the column 3 5 0 −4 1 6 ↓ 15 5 15
  6. 6. Step #6: Multiply the sum, 15, by r; 15g 3=15, and place this number under the next coefficient, then add the column again. 3 Add 5 0 −4 1 6 ↓ 15 45 5 15 41 Multiply the diagonals, add the columns.
  7. 7. Step #7: Repeat the same procedure as step #6. 3 5 Add Columns 0 Add Columns −4 1 Add Columns Add Columns 6 ↓ 15 45 123 372 5 15 41 124 378
  8. 8. Step #8: Write the quotient. The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.
  9. 9. The quotient is: 378 5x + 15x + 41x + 124 + x−3 3 2 Remember to place the remainder over the divisor.
  10. 10. Try this one: 1) (t 3 − 6t 2 + 1) ÷ ( t + 2) −2 1 −6 0 1 −2 16 −32 1 −8 16 −31 31 Quotient = 1t − 8t + 16 − t+2 2

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