+73 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)
+128406 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 !!!
+128406 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 !!!!
