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For each problem below, find the quotient and remainder. Then, use the remainder theorem to
check your remainder. 

 

a.) (x^4 + 10x + 5)/(x + 2)

 

b.) (2x^3 - 5x^2 + 3x - 1)/(x - 3)

 Feb 16, 2019
 #1
avatar+104836 
+2

a.) (x^4 + 10x + 5)/(x + 2)

 

Gotta' be careful to account for missing powers, GM.....so we have

 

 

              x^3                   - 2x^2    +   4x    + 2

x + 2   [  x^4    + 0 x^3   +  0x^2  +  10x   + 5 ]

              x^4    + 2x^3

           ________________________________

                         -2x^3  + 0 x ^2

                        -2x^3  -  4x^2

                      ___________________________

                                    4x^2   + 10x

                                    4x^2 +    8x

                                   _____________________

                                                2x     +    5

                                                2x     +   4

                                               _____________

                                                              1

 

Checking the Remainder Theorem  (remember that x + 2 is the divisor....so we are checking  x + 2 = 0 ⇒ x = -2  )

 

(-2)^4 + 10 (-2) + 5

 

16 - 20 + 5

 

21 - 20  =

 

1

 

Yep......works right out    !!!

 

 

cool cool cool

 Feb 16, 2019
 #2
avatar+104836 
+2

b.) (2x^3 - 5x^2 + 3x - 1)/(x - 3)

 

Since we don't have any missing powers in the dividend polynomial......I'm going to use synthetic division to find the residual polynomial and remainder

 

The divisor is     x - 3 = 0  ⇒   x = 3

 

So we have

 

 

3 [    2        - 5         3          -1 ]

                    6         3         18

       _____________________

       2           1        6         17

 

The remaining polynomial (quotient)  is    2x^2   + x   + 6

 

The remainder is  17

 

Checking  x = 3    with the Remainder Theorem...we get

 

2(3)^3  - 5(3)^2   + 3(3)  - 1   =

 

54  - 45 + 9 - 1   =

 

63   -  46  =

 

17......just as we hoped for   !!!!

 

 

cool cool cool

 Feb 16, 2019

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