(x^3-4x+6)/(x+3) Synthetic Division

Jasonbradley

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Jun 17, 2024

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Performing Synthetic Division: (x³ - 4x + 6) / (x + 3)

Synthetic division is a shortcut method for dividing polynomials, especially when the divisor is a linear expression of the form (x - a). Let's apply this method to divide (x³ - 4x + 6) by (x + 3).

Setting Up the Problem

1. Identify the coefficients: In our dividend (x³ - 4x + 6), the coefficients are 1, 0, -4, and 6. Notice we need a placeholder for the missing x² term.
2. Identify the divisor: Our divisor is (x + 3), so 'a' is -3.

Now, let's set up the synthetic division problem:

 -3 | 1  0  -4  6
     _________________

Performing the Calculation

1. Bring down the first coefficient: Bring down the '1' below the line.

 -3 | 1  0  -4  6
     _________________
       1 

2. Multiply and add: Multiply the '1' by -3 (the divisor) and write the result (-3) below the '0'. Add the numbers in the second column (0 + (-3) = -3).

 -3 | 1  0  -4  6
     _________________
       1  -3

3. Repeat the process: Multiply the -3 by -3, and write the result (9) below the -4. Add the numbers in the third column (-4 + 9 = 5).

 -3 | 1  0  -4  6
     _________________
       1  -3   5

4. Final step: Multiply the 5 by -3, and write the result (-15) below the 6. Add the numbers in the last column (6 + (-15) = -9).

 -3 | 1  0  -4  6
     _________________
       1  -3   5  -9

Interpreting the Result

The numbers below the line (1, -3, 5, and -9) represent the coefficients of the quotient and the remainder. Starting from the left, the coefficients correspond to the powers of x, decreasing by one each time.

• Quotient: x² - 3x + 5
• Remainder: -9

Therefore, the result of the division (x³ - 4x + 6) / (x + 3) is:

(x³ - 4x + 6) / (x + 3) = x² - 3x + 5 - 9/(x + 3)

Note: The remainder is expressed as a fraction with the divisor as the denominator.