(x 4 - 81) (x - 3)

Question

use synthetic division 

Answer 1

(x⁴ - 81) / (x - 3)

 

 

When using synthetic division, we divide the coefficients of the polynomial by the root of its factor.

 

The coefficient of the polynomial (x⁴ - 81)  is   1  0  0  0  -81

The root of the factor  (x - 3)   is  3

 

 

3 | 1    0    0     0     -81

 

 

______________________

 

 

 

Bring down the 1 under the line.

 

 

3 | 1    0    0     0     -81
 
 
______________________

     1

 

 

Multiply the 1 by the root.  Place its product as the addend in the next column.

 

 

 

3 | 1    0    0     0     -81
 
           3
______________________

     1

 

 

 

Add the numbers in the second column.

 

 

3 | 1    0    0     0     -81
 
           3
______________________

     1    3

 

 

Multiply the sum by the root 3.  Place the product as  the addend in the next column.

 

 

 

3 | 1    0    0     0     -81
 
           3    9
______________________

     1    3    9

 

 

 

Repeat these steps for each column.

 

 

 

 

3 | 1    0    0     0    -81
 
           3    9    27    81
______________________
     1    3    9    27     0

 

 

The digits under this line is the quotient of the polynomial.  The last digit in the quotient is the remainder.  Since the remainder in this division is zero, the quotient is a factor of the polynomial.

 

The quotient is:   x³ + 3x² + 9x + 27