(4x2−13x−5)÷(x−2) Synthetic Division

Jasonbradley

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Jun 16, 2024

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Using Synthetic Division to Divide (4x² - 13x - 5) by (x - 2)

Synthetic division is a shortcut method for dividing polynomials, particularly when the divisor is in the form (x - a). It simplifies the long division process, making it easier to find the quotient and remainder. Let's see how to use synthetic division to divide (4x² - 13x - 5) by (x - 2).

Steps:

1. Set up the synthetic division:

   • Write the coefficients of the dividend (4x² - 13x - 5) in a row: 4 -13 -5
   • Write the value of 'a' from the divisor (x - 2) to the left of the coefficients: 2 | 4 -13 -5
   • Draw a line below the coefficients.

2. Bring down the leading coefficient:

   • Bring down the first coefficient (4) below the line: 2 | 4 -13 -5 ** 4**

3. Multiply and add:

   • Multiply the 'a' value (2) by the number you just brought down (4) and write the result (8) under the next coefficient (-13). 2 | 4 -13 -5 ** 4 8**
   • Add the numbers in the second column (-13 + 8 = -5) and write the sum below the line. 2 | 4 -13 -5 ** 4 8** ** 4 -5**

4. Repeat the process:

   • Multiply the 'a' value (2) by the new number (-5) and write the result (-10) under the next coefficient (-5). 2 | 4 -13 -5 ** 4 8 -10** ** 4 -5**
   • Add the numbers in the last column (-5 - 10 = -15) and write the sum below the line. 2 | 4 -13 -5 ** 4 8 -10** ** 4 -5 -15**

Interpreting the Results:

• The numbers below the line (4, -5) represent the coefficients of the quotient polynomial.
• The last number (-15) is the remainder.

Therefore, the result of dividing (4x² - 13x - 5) by (x - 2) is:

(4x² - 13x - 5) ÷ (x - 2) = 4x - 5 - 15/(x - 2)

In conclusion, synthetic division provides a convenient way to perform polynomial division, especially when the divisor is linear. By following the simple steps, you can efficiently find the quotient and remainder of the division.