(x^3-2x^2-5x+6)/(x-1) Synthetic Division

Jasonbradley

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Jun 17, 2024

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Using Synthetic Division to Divide (x^3 - 2x^2 - 5x + 6) by (x - 1)

Synthetic division is a shortcut method for dividing polynomials, especially when the divisor is in the form of (x - a). Let's illustrate this with the example of dividing (x^3 - 2x^2 - 5x + 6) by (x - 1).

Step 1: Set up the Synthetic Division

1. Write down the coefficients of the dividend (the polynomial being divided), including any zero coefficients for missing terms. In this case, we have: 1 -2 -5 6

2. Write the constant term of the divisor (x - 1) with the opposite sign, which is 1, to the left of the coefficients.

   1 | 1 -2 -5 6

Step 2: Perform the Synthetic Division

1. Bring down the first coefficient (1) below the line.

   1 | 1 -2 -5 6 ------------- 1

2. Multiply the number you just brought down (1) by the divisor (1), and write the result (1) below the next coefficient (-2).

   1 | 1 -2 -5 6 ------------- 1 1

3. Add the two numbers in the second column (-2 and 1), and write the result (-1) below the line.

   1 | 1 -2 -5 6 ------------- 1 1 -1

4. Repeat steps 2 and 3 for the remaining coefficients.

   1 | 1 -2 -5 6 ------------- 1 1 -4 -1 -4 -9

Step 3: Interpret the Results

1. The numbers below the line, excluding the last one, represent the coefficients of the quotient. In this case, the quotient is x^2 - x - 4.
2. The last number below the line (-9) is the remainder.

Therefore, the result of dividing (x^3 - 2x^2 - 5x + 6) by (x - 1) is:

x^2 - x - 4 - 9/(x - 1)

Advantages of Synthetic Division

Synthetic division provides a more compact and efficient method compared to long division for polynomial division. It simplifies the process and reduces the chances of calculation errors.