I kind of know synthetic division but the (a) is throwing me off a bit? Can you help?

Question

Given P(x) = 6x^5 + 13x^4 + ax^3 - 21x^2 - 3a + 16 and (x+2) is a factor of P(x), us synthetic division to find a.
Can anyone explain this or share possible theroms or bolth. Please Help!!!!!!!!!

Answer 1

If you make a synthetic division table (it's not ideal here since Yahoo cuts it off)

note: I'm assuming the last term is -3a+16 and its not -3ax + 16

-2 | 6....13..........a..........-21...............0...............-3a+16
............-12.........-2.......(-2a+4)....(4a+34)..........-8a-68
---------------------------------------------------------------
........6....1........(a-2).....(-2a-17)....(4a+34)......(-11a-52)


Our last column (remainder column) looks like this:
-3a+16
-8a-68
------------
-11a-52


Since your remainder is 0 we can say

-11a-52=0

So solving for a gives us:

-11a=52
a=-52/11

So a= -52/11 or a=-4.72727272727272

Answer 2

Are you missing an "x" after the "-3a" term? I assume you mean:

P(x) = 6x^5 + 13x^4 + ax^3 - 21x^2 - 3ax + 16

We have:

-2 | 6 ... 13 ...... a ...... -21 ...... -3a ........ 16
.... |___ -12__ -2 __-2a+4__4a+34 __-2a-68

...... 6 .....1 ....a-2 .... -2a-17 .... a+34 ....-2a-52

For this to factor the remainder term must equal zero. Therefore:

-2a - 52 = 0
2a = -52
a = -26

So we have:

P(x) = 6x^5 + 13x^4 + ax³ - 21x² - 3ax + 16
P(x) = 6x^5 + 13x^4 - 26x³ - 21x² + 78x + 16

P(x) = (x + 2)[6x^4 + x³ + (a - 2)x² + (-2a - 17)x + (a + 34)]
P(x) = (x + 2)(6x^4 + x³ - 28x² + 35x + 8)