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# For each problem below, find the quotient and remainder. Please Help! I do I check the remainder like this?

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

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

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

Feb 16, 2019

#1
+111328
+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    !!!

Feb 16, 2019
#2
+111328
+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   !!!!

Feb 16, 2019