(x^3-7x-6)/(x-2) Synthetic Division

Jasonbradley

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Jun 17, 2024

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Performing Synthetic Division: (x³ - 7x - 6) / (x - 2)

Synthetic division is a simplified method for dividing polynomials, particularly useful when the divisor is of the form (x - a). Let's walk through the steps to divide (x³ - 7x - 6) by (x - 2) using synthetic division.

Step 1: Set Up the Problem

1. Identify the coefficients: Write down the coefficients of the dividend (x³ - 7x - 6), remembering to include a '0' for the missing x² term: 1, 0, -7, -6.
2. Identify the constant term of the divisor: The constant term in (x - 2) is -2.

Now, set up the synthetic division table:

 -2 | 1   0   -7   -6 
      -----------------

Step 2: Bring Down the First Coefficient

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

 -2 | 1   0   -7   -6 
      -----------------
        1 

Step 3: Multiply and Add

1. Multiply: Multiply the number you just brought down (1) by the constant term of the divisor (-2) and write the result ( -2) below the second coefficient.

 -2 | 1   0   -7   -6 
      -----------------
        1   -2

2. Add: Add the second coefficient (0) and the result (-2) you just wrote. Place the sum (-2) below the line.

 -2 | 1   0   -7   -6 
      -----------------
        1   -2

Step 4: Repeat Steps 3 and 4

1. Multiply: Multiply the new number below the line (-2) by the constant term of the divisor (-2) and write the result (4) below the third coefficient.

 -2 | 1   0   -7   -6 
      -----------------
        1   -2    4

2. Add: Add the third coefficient (-7) and the result (4) you just wrote. Place the sum (-3) below the line.

 -2 | 1   0   -7   -6 
      -----------------
        1   -2    4   -3

Step 5: Final Multiplication and Addition

1. Multiply: Multiply the last number below the line (-3) by the constant term of the divisor (-2) and write the result (6) below the last coefficient.

 -2 | 1   0   -7   -6 
      -----------------
        1   -2    4   -3    6

2. Add: Add the last coefficient (-6) and the result (6) you just wrote. Place the sum (0) below the line. This is our remainder.

 -2 | 1   0   -7   -6 
      -----------------
        1   -2    4   -3    0

Step 6: Interpret the Result

The numbers below the line (1, -2, 4, -3) represent the coefficients of the quotient. Since the remainder is 0, we can write the result as:

x³ - 7x - 6 = (x - 2)(x² - 2x + 4) + 0

Therefore, (x³ - 7x - 6) / (x - 2) = x² - 2x + 4.