MHB [ASK] polynomial f(x) divided by (x - 1)

[Monoxdifly]

A polynomial f(x) = [math]2x^3-5x^2+ax+18[/math] is divisible by (x - 3). The result of that polynomial f(x) divided by (x - 1) is ...
A. [math]2x^2-7x+2[/math]
B. [math]2x^2+7x-2[/math]
C. [math]2x^2-7x-2[/math]
D. [math]x^2-6x-2[/math]
E. [math]x^2-6x+3[/math]

I got a + 3 = -6 and so a = -9 and f(x) = [math]2x^3-5x^2-9x+18[/math], but when I divided it with x - 1 I got [math]x^2-3x+6[/math] with the remainder 24. I'm quite sure that the question isn't wrong since it's in a national level exam, but where did I go wrong?

 

[skeeter]

A polynomial f(x) = [math]2x^3-5x^2+ax+18[/math] is divisible by (x - 3).

If a polynomial is divisible by $(x-k)$, then $f(k) = 0$

$f(3) = 0 \implies a = -9 \implies f(1) = 6$

$f(1) \ne 0 \implies f(x)$ is not divisible by $(x-1)$.

My opinion is there is an error in the question somewhere.

 

[tkhunny]

Synthetic Division does some useful things.

3 |  2  -5    a       18
         6    3     3a + 9
-------------------------
     2   1  a+3      3a+27 = 0 ==> a = -9

1 |  2  -5    -9      18
         2    -3     -12
-------------------------
     2  -3   -12      6

And thus we see: $2x^{2} - 3x - 12 + \dfrac{6}{x-1}$

How did you get only x^2, instead of 2x^2? Something wrong there.