(2x^2+x-7)/(x-5) Synthetic Division

Jasonbradley

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Jun 16, 2024

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Synthetic Division: (2x^2 + x - 7) / (x - 5)

Synthetic division is a simplified method for dividing polynomials, particularly when the divisor is in the form (x - a). It streamlines the long division process, making it faster and less prone to errors. Let's demonstrate how to use synthetic division to divide (2x^2 + x - 7) by (x - 5).

Steps for Synthetic Division

1. Set up the problem:

   • Write down the coefficients of the dividend (2x^2 + x - 7): 2, 1, -7
   • Write down the value of 'a' from the divisor (x - 5): 5

       5 | 2  1  -7 
           ----------------
   

2. Bring down the leading coefficient:

   • Bring down the first coefficient (2) below the line.

       5 | 2  1  -7 
           ----------------
           2 
   

3. Multiply and add:

   • Multiply the number below the line (2) by the divisor's value (5): 2 * 5 = 10
   • Write the result (10) above the line in the next column.
   • Add the numbers in the second column: 1 + 10 = 11
   • Write the sum (11) below the line.

       5 | 2  1  -7 
           ----------------
           2  11 
   

4. Repeat steps 3 and 4:

   • Multiply the new number below the line (11) by the divisor's value (5): 11 * 5 = 55
   • Write the result (55) above the line in the next column.
   • Add the numbers in the third column: -7 + 55 = 48
   • Write the sum (48) below the line.

       5 | 2  1  -7 
           ----------------
           2  11 48 
   

5. Interpret the results:

   • The numbers below the line, except for the last one, represent the coefficients of the quotient.
   • The last number below the line is the remainder.

   Therefore, the quotient is 2x + 11 and the remainder is 48.

Writing the Complete Solution

The result of the synthetic division can be expressed as:

(2x^2 + x - 7) / (x - 5) = 2x + 11 + 48/(x - 5)

This means that the original polynomial (2x^2 + x - 7) can be written as the product of the divisor (x - 5) and the quotient (2x + 11), plus the remainder (48).

Synthetic division provides a concise and efficient way to divide polynomials. By following these steps, you can perform the division quickly and accurately, making it a valuable tool for polynomial operations.