(x^3-4x^2+6x-4)/(x-2) Synthetic Division

Jasonbradley

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Jun 17, 2024

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Solving (x³ - 4x² + 6x - 4) / (x - 2) using Synthetic Division

Synthetic division is a shortcut method for dividing polynomials, particularly when the divisor is a linear expression in the form of (x - a). Let's walk through how to solve (x³ - 4x² + 6x - 4) / (x - 2) using synthetic division.

Steps for Synthetic Division

1. Set up the problem:

   • Write the coefficients of the dividend (x³ - 4x² + 6x - 4) in a row: 1 -4 6 -4
   • Write the value of 'a' from the divisor (x - 2) to the left of the coefficients: 2 | 1 -4 6 -4

2. Bring down the leading coefficient:

   • Bring down the first coefficient (1) below the line. 2 | 1 -4 6 -4
   • 1

3. Multiply and add:

   • Multiply the value 'a' (2) by the number you just brought down (1): 2 * 1 = 2
   • Add the result (2) to the next coefficient (-4): -4 + 2 = -2
   • Write this result (-2) below the line: 2 | 1 -4 6 -4
   • 1 -2

4. Repeat steps 3 and 4:

   • Multiply the value 'a' (2) by the last number (-2): 2 * -2 = -4

   • Add the result (-4) to the next coefficient (6): 6 - 4 = 2

   • Write this result (2) below the line: 2 | 1 -4 6 -4

   • 1 -2 2

   • Multiply the value 'a' (2) by the last number (2): 2 * 2 = 4

   • Add the result (4) to the next coefficient (-4): -4 + 4 = 0

   • Write this result (0) below the line: 2 | 1 -4 6 -4

   • 1 -2 2 0

5. Interpret the results:

   • The last number (0) is the remainder.
   • The other numbers (1 -2 2) are the coefficients of the quotient, starting with the highest power of x (x²).

The Solution

Therefore, the quotient is x² - 2x + 2 and the remainder is 0. This means that (x³ - 4x² + 6x - 4) is perfectly divisible by (x - 2), and the result is x² - 2x + 2.