how to solve this synthetic division problem n^5+11n^4+24n^3-24n^2-44n-41/n+6

Question

i want to know how to sovle this synthetic dividion problem. 

Answer 1

Hi Hunter, synthetic division is easy enough if you remember the set up. 

 

The set up:

 

The denominator must be linear. That means the power of the variable must be 1. If it is not, synthetic division cannot be used.

 

The coefficient of the variable in the denominator must be 1. If it is not, divide the denominator by the coefficient to make it one.

 

Example: denominator is 4n + 5. We need to divide by 4 to get n + 5/4. The coefficient on the n is now 1.

 

For our problem, set the denominator, n + 6, equal to zero to find out what we will be dividing by.

 

n + 6 = 0

 

Subtract 6 from both sides giving:

 

n = -6

 

We only work with the coefficients of the polynomial. Place the divisor to the left, and the coefficients of the polynomial to the right. If there are missing powers in the polynomial, use a 0 as a place holder for that power.

 

Example: The polynomial is: x³ + 3x -1, The coefficients used for division would be 1, 0, 3, 1. The 0 represent the missing x² term.

 

Our coefficients:

 

-6 | 1    11    24    -24    -44    -41

    |

    |      

    +---------------------------------

 

Bring down the 1 and place it below the dashed line.

Multiply the 1 by -6 and place the results under the second number.

Add the two numbers together and place the result under the dashed line.

Multiply the 5 by -6 and place the results under the third number.

Continue with this adding and multiplying until all of the numbers have been used.

 

-6 | 1    11    24    -24    -44    -41
    |
    |       -5
   +---------------------------------

      1      6

 

The completed division:

 

-6 | 1    11    24    -24    -44    -41
    |
    |       -6   -30     36    -72    696
    +---------------------------------
      1      5    -6     12   -116    645

 

The division gives:

 

Remember, the results of the division will start at one power less than the original. In our problem, the first power was 5. In the results, the first power will be 4.

 

n⁴ + 5n³ - 6n² + 12n - 116 with a remainder of 645 / (n + 6)

 

The result of the division:

 

Since the division does not end with a zero under the last number, and there is a remainder, n + 6 is not a factor of the given polynomial.

 

Questions? 

Answer 2

n+6 = n-(-6)

 

-6⌋     1          11         24         -24         -44         -41

     _________-6_____-30_____36_____-72_____696

          1           5         -6           12        -116        655

 

So, (n⁵+11n⁴+24n³-24n²-44n-41)/(n+6) =                                   

 

n⁴+5n³-6n²+12n-116 with a remainder of 655