solve 4x^2-6x+3/x-3

Question

solve quotient by long division

Answer 1

 

         4x + 6          

x-3 | 4x² - 6x + 3

     - (4x² -12x)

                 6x + 3

               -(6x -18)

                       21

 

Remainder is 21.

 

(4x² -6x + 3) / (x-3) = 4x + 6 + (21/(x-3))

 

 

Check:

x = 5

 

(4(5)² -6(5) + 3)   / 5-3

 

(4(25)-30+3) / 2

 

73/2 = 36 1/2

 

 

4(5) + 6  + (21/5-3)

 

20 + 6 + 21/2

 

20 + 6 + 10 1/2

 

36 1/2 ... check.

 

Answer 2

 This is a problem of division of 2 polynomials.

 

 

    ( 4X² - 6X + 3 ) / ( X - 3 )        It is important to place parenthesis.

 

    first try to factor the numerator, to see if it has a common factor of ( X -3) .

 

      4X² - 6X + 3  is not factorable , There is no 2 integer whose difference is 6 ,and product 12.

 

       Have to carry long division:

 

                4X +6

       X - 3 l  4X² - 6X + 3

                 4X² - 12X

                      + 12X

                      6X + 3

                      6X - 18

                          + 18 

                           21

 

 So 4X² - 6X + 3 = ( X- 3) ( 4X + 6 ) + 21  ( 1)

 

     Just like  17 ÷ 4 = 4 , remainder of 1

 

                  17 = 4* 4 +1

 

             The idea is the same as in arithmetic when you convert mixed numbers to improper fractions, and

             vice versa.

        From equation (1) , you can see that

       

      f(X) = 4X² - 6X + 3 = ( X -3) (4X+ 6) +21

 

               f(3) = 21

        This idea becomes very useful in calculus, by using Chinese remainder theorem.

        So, it is important to keep focus on it.