(x^4-8)÷(x+3)

Question

Long Division of polynomials 

Answer 1

You may learn synthetic division much later.  Long division (you learned it in grade school) is tedious, but here goes (be sure to include the zeros):

                     x^3
                -------------------------------------------------
       x+3  )   x^4   +0*X^3  + 0*s^2  + 0*x  - 8
                   x^4  + 3*x^3
                     -------------------
                            - 3*x^3   - 0*x^2

 

then,

                    x^3  - 3x^2
                -------------------------------------------------
       x+3  )   x^4   +0*X^3  + 0*s^2  + 0*x  - 8
                   x^4  + 3x^3
                 -------------------
                             - 3x^3   - 0*x^2
                             - 3x^3   - 9x^2
                             -------------------------
                                            9x^2    +   0*x
 
then,

                    x^3  - 3x^2  +  9x
                -------------------------------------------------
       x+3  )   x^4   +0*X^3  + 0*s^2  + 0*x  - 8
                   x^4  + 3x^3
                -------------------
                            - 3x^3   - 0*x^2
                            - 3x^3   - 9x^2
                        -------------------------
                                          9x^2    +   0*x
                                          9x^2    +  27x
                                        --------------------
                                                         -27 x     - 8

 
then,

 

                    x^3   -  3x^2    +   9x  -  27
                -------------------------------------------------
       x+3  )   x^4   +0*X^3  + 0*s^2  + 0*x  - 8
                   x^4  + 3x^3
                -------------------
                            - 3x^3   - 0*x^2
                            - 3x^3   - 9x^2
                        -------------------------
                                           9x^2    +   0*x
                                           9x^2    + 27x
                                        --------------------
                                                         -27 x    -   8
                                                         -27x    -  81
                                                       -----------------
                                                                       73

So, the answer is  x^3 -3x^2 + 9x -27 with remainder of 73

        Often the remainder is written as a fraction:    x^3 -3x^2 + 9x -27 + 73/(x+3)

Answer 2

Hi Precious,

 

One way that simple polynomial division problems like this (where a particular "zero" is being divided out) can be solved is known as synthetic division, and it involves manipulating the coefficients of the polynomial in a clever way to generate the answer to the division problem.

 

We start by writing down the coefficients

x⁴  1

x³  0

x²  0

x¹  0

x⁰ -8

 

We then place the "zero" -- the value of x which makes the divisor zero (here x + 3 = 0 → x = -3) at the top of the chart:

-3

 1

 0

 0

 0

-8

 

The first coefficient in the chart will stay unchanged, but the other coefficients will need to be updated.

 

First, multiply the first coefficient in the chart by the "zero" and add the product to the second coefficient to get a new second coefficient:

-3
 1
 0 + 1×-3 = -3
 0
 0
-8

 

To get the third coefficient, multiply the new second coefficient by the "zero" and add the product to the third coefficient:

-3
 1 
 0 + 1×-3 = -3
 0 + -3×-3 = 9
 0 
-8

 

Continue doing this until you have updated all of the coefficients:

-3
 1
 0 +    1×-3 = -3
 0 +   -3×-3 =  9
 0 +    9×-3 = -27
-8 + -27×-3 = 73

 

This new column of coefficients are for the resulting polynomial:

x³   1

x²  -3

x¹   9

x⁰ -27

remainder 73

 

So the answer is x³ - 3x² + 9x - 27 + 73/(x+3)

 

Answer 3

the answer would be x³-3x²+9x-27 with a remainder of 73.