factorize please

Question

-x³+5x²-5x+3

Answer 1

Very astute of the previous tutor to notice the lack of the coefficient (3) in the original equation. The above answer is perfect, however, there is no explanation why each step was carried out.  I would just like to elaborate some for the student.  Thank you and I hope it helps.

 

1)  -3x³  + 5x² - 5x +3

      Pull the coefficient and x from the term 5x² - 5x = 5x(x-1)

 

Now one has:

 

     -3x³ + 5x(x-1) +3

 

2)  Combine -3x³ and +3 and take out the coefficient 3 to get

 

     -3(x³+1)+ 5x(x-1)

 

3)  (x³ +1) can be further factored into (x² +x+1)(x-1)

 

      By multiplying the two terms using longhand arithmetic, one can see that this is true:

 

               x²+  x +  1

                      x  -  1

                -  x² -  x  -1

            x³ + x²+ x___

            x³                     -1 → (x³ -1)

 

4) Now, insert these further factored terms back into the equation to get

 

   [-3(x-1)(x² + x +1)] + [5x(x-1)]

 

Note:  I have bracketed the two sections above showing that (x-1) is the common multiplier between the 2 bracketed terms showing that the equation can then be further simplified:

 

a.  First multiply -3(x² + x + 1) = (-3x²  - 3x -3) getting:

 

      (x-1)(-3x² -3x -3) + 5x(x-1)

 

b. Then, multiply all by (x-1)

 

       (x-1)(-3x² -3x +5x -3) = (x-1)(-3x² +2x -3) 

    

 

Answer 2

Seems like a coefficient of 3 is missing from your cubic term.

 

 - 3 X^3 + 5 X^2 - 5x + 3 =

 

- 3 ( X^3 - 1) + 5X ( X - 1 ) =

 

   -3 ( X -1 ) ( X^2 + X + 1 ) + 5X ( X - 1 ) =

 

    ( X - 1 ) ( -3X^2 - 3X - 3 + 5X)  =

 

    ( X - 1 ) ( - 3 X^2 + 2X - 3 )