if the the divisor have exponent

Question

  x⁵  - 5x⁴ + 7x³ + x² - 8x + 7 divide x²- 2

Answer 1

(x^5 - 5x^4 + 7x^3 + x^2 - 8x + 7) / (x^2 - 2)

The easiest way is to use long division:

              x^3 - 5x^2 + 9x - 9                         
x^2 - 2 | x^5 - 5x^4 + 7x^3 + x^2   - 8x  + 7
              x^5             - 2x^3
                0   - 5x^4 + 9x^3
                     - 5x^4          + 10x^2
                         0     + 9x^3 - 9x^2
                                + 9x^3            - 18x
                                     0     - 9x^2 + 10x
                                            - 9x^2          + 18
                                                0     + 10x  - 11

check:

(x^2 - 2)(x^3 - 5x^2 + 9x - 9) + (10x - 11)

=? x^5 - 5x^4 + 7x^3 + x^2 - 8x + 7

x^2(x^3 - 5x^2 + 9x - 9)
   -2(x^3 - 5x^2 + 9x - 9) + (10x - 11)

x^5 - 5x^4 + 9x^3   - 9x^2
                   - 2x^3 + 10x^2 - 18x + 18
                                           + 10x - 11

x^5 - 5x^4 + 7x^3 + x^2 - 8x + 7

= x^5 - 5x^4 + 7x^3 + x^2 - 8x + 7 checked!

 

I also uploaded a graphical check here:

http://www.wyzant.com/resources/files/260020/checking_long_division_with_geogebra

Answer 2

Hi Alliv,

 

             x³-5x²+9x-9

x²-2    ⁄ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

            x⁵-5x⁴+7x³+x²-8x+7

       -    x⁵       -2x³                         (when we         subtract the two equations the signs of the bottom equation changes)

               -5x⁴+9x³+x²                      

         -    -5x⁴       +10x²                   

                       9x³-9x²-8x

                 -      9x³      -18x

                            -9x²+10x+7

                       -     -9x²       +18

                                   +10x-11      Remainder

 

Check:  (x²-2)(x³-5x²+9x-9)+(10x-11)=x⁵-5x⁴+9x³-9x²-2x³+10x²-18x+18+10x-11

                                                          =x⁵-5x⁴+7x³+x²-8x+7  

Answer 3

To answer this question you must use long division. You ask yourself, "what multiplies with x^2 to get x^5?" The answer is x^3. So that is the first part of your answer. Now multiply x^3 by (x^2-2). This gives you x^5-2x^2. You subtract this from x^5-5x^4+7x^3+x^2-8x+7, leaving you with -5x^4+9x^3+x^2-8x+7. Now you ask yourself, "what multiplies with x^2 to get -5x^4?" The answer is -5x^2. So that is now the second term in your answer. Now multiply -5x^4 by (x^2-2). This gives you -5x^4+2x^2. You subtract this from -5x^4+9x^3+x^2-8x+7, leaving you with 9x^3-x^2-8x+7. Continue this process until you can no longer divide. Then your remainder is added at the end as a fraction (with the denominator as x^2-2).

 

Final answer:

 

x^3-5x^2+9x-1+(10x+5)/(x^2-2)

 

 

To check this answer, you multiply (x^2-2)[x^3-5x^2+9x-1+(10x+5)/(x^2-2)] which will give you x^5-5x^4+7x^3+x^2-8x+7.