Question

Divide 6x^6+5x^5+2x^4-9x^3+7x^2-10x+2 by 3x + 1 by using long division. Show all work and steps work. Then explain if 3x + 1 is a factor of the dividend.

Answer 1

Answer:

1)   

  Quotient: 2x⁵ + x⁴ + (1/3)x³ -(28/9)x² + (91/27)x – (361/81)

  Remainder: 523/81

2) Since the division is not exact (remainder 523/81) the factor 3x + 1 (the divisor) is not a factor.

Explanation (Long division):

1)

6x⁶ + 5x⁵ + 2x⁴ - 9x³ + 7x² - 10x + 2  | 3x + 1
                                                         --------------------------------------------
                                                           2x⁵

2)

  6x⁶ + 5x⁵ + 2x⁴ - 9x³ + 7x² - 10x + 2  | 3x + 1
                                                           --------------------------------------------
-6x⁶ - 2x⁵                                               2x⁵
-------------------------------------------------
          3x⁵ + 2x⁴ - 9x³ + 7x² - 10x + 2

3)

  6x⁶ + 5x⁵ + 2x⁴ - 9x³ + 7x² - 10x + 2  | 3x + 1
                                                           --------------------------------------------
-6x⁶ - 2x⁵                                               2x⁵ + x⁴
-------------------------------------------------
          3x⁵ + 2x⁴ - 9x³ + 7x² - 10x + 2


4) 

 6x⁶ + 5x⁵ + 2x⁴ - 9x³ + 7x² - 10x + 2  | 3x + 1
                                                           --------------------
-6x⁶ - 2x⁵                                               2x⁵ + x⁴
-------------------------------------------------
          3x⁵ + 2x⁴ - 9x³ + 7x² - 10x + 2
        - 3x⁵ - x⁴
--------------------------------------------------
                 2x⁴ - 9x³ + 7x² - 10x + 2


Since the number of characters to show here is limited, I continue with this division is a separate file.

Please, open the attached pdf file. There you have the entire division.

Remember that if 3x + 1 were a factor the division would be exact. So, since the division is not exact 3x + 1 is not a factor.