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

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

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

Bring down the first coefficient (1) below the line.
1  1 2 5 6  1

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

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

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
 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.
 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.