2x^3+5x^2+4=(x+3)Q(x)+R

Question

find the polynomial Q(x) and the constant R

Answer 1

find the polynomial Q(x) and the constant R

f(x) = 2x^3+5x^2+4=(x+3)Q(x)+R

 

Using Synthetic Division/Substitution and the Remainder Theorem:

 

-3 | 2  5 0   4

----    -6 3  -9

      ------------

      2 -1 3 | -5

                ----

Q(x) = 2x^2 - x + 3 and R = -5 = f(-3)

 

Check: (x+3)(2x^2 - x + 3)-5

= 2x^3 - x^2 + 3x + 6x^2 - 3x + 9 - 5

= 2x^3 + 5x^2 + 4

= f(x) √

Answer 2

 

               2X^2 -  X  + 3

                                           

     X +3 l 2 X^3 + 5 X^2 + 4

               2X ^3 ± 6 X^2

                          - X^2 +4

                          -X^2 -3X                                                                             +3X +4

                                    - 3X - 9 

                                           -5

    2 X^3 + 5 X^2 + 4 = ( X + 3 ) ( 2X^2 - X - 3 ) + R

 

          Q( X ) = 2X^2 - X + 3      R = -5