Use synthetic division to determine the quotient and remainder.

Question

(x3 − 41x − 30) ÷ (x + 6)

Answer 1

The process is the divide the coefficients (descending order of variable) of the dividend by the possible root (divisor).

 

Since -6 is your possible root, we set up the division as this.

 

 

-6 | 1       0         -41         -30

 

              -6          36          30

____________________________

 

      1      -6          -5            0

 

 

As you can see, the digits below the line is the quotient.  The last digit under the line is the remainder. 

 

Quotient = x² - 6x - 5

Remainder = 0

 

If the reminder is zero, then the quotient is a factor of the dividend.  In this case, we have a remainder of zero.  Therefore,

 

(x² - 6x - 5) is a factor of (x³ - 41x - 30).

 

Answer 2

The key value for synthetic division is x + 6 = 0 or x = -6

 

-6|   1     0    -41    -30

              -6    36     30

        1   -6     -5    | 0

 

The bottom line shows the coefficients of the answer with remainder zero

 

Thus (x³ - 41x - 30)/(x + 6) = x² - 6x - 5