%2F(x%2B2)-synthetic-division.jpeg "(x^3+5x^2-x-9)/(x+2) Synthetic Division")
Synthetic Division: A Step-by-Step Guide to Dividing (x³ + 5x² - x - 9) by (x + 2)
Synthetic division is a shortcut method for dividing polynomials, especially when the divisor is a linear expression of the form (x - a). In this case, we'll divide (x³ + 5x² - x - 9) by (x + 2).
Step 1: Set up the problem
- Write the coefficients of the dividend (x³ + 5x² - x - 9) in a row: 1 5 -1 -9.
- Write the constant term of the divisor (x + 2) with its sign changed: -2.
- Draw a horizontal line below the coefficients.
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-2 | 1 5 -1 -9
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Step 2: Bring down the first coefficient
- Bring down the first coefficient (1) below the line.
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-2 | 1 5 -1 -9
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1
Step 3: Multiply and add
- Multiply the number you just brought down (1) by the divisor's constant term (-2) and write the product ( -2 ) under the next coefficient (5).
- Add the numbers in the column (5 + -2 = 3).
_________________
-2 | 1 5 -1 -9
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1 3
Step 4: Repeat the multiplication and addition process
- Multiply the new number (3) by the divisor's constant term (-2) and write the product (-6) under the next coefficient (-1).
- Add the numbers in the column (-1 + -6 = -7).
_________________
-2 | 1 5 -1 -9
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1 3 -7
Step 5: Continue the process until you reach the last coefficient
- Multiply the new number (-7) by the divisor's constant term (-2) and write the product (14) under the last coefficient (-9).
- Add the numbers in the column (-9 + 14 = 5).
_________________
-2 | 1 5 -1 -9
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1 3 -7 5
Step 6: Interpret the results
- The numbers below the line represent the coefficients of the quotient, starting from the highest power of x.
- The last number (5) represents the remainder.
Therefore, the result of the division (x³ + 5x² - x - 9) / (x + 2) is:
x² + 3x - 7 + 5/(x + 2)