What is the value of k if x3 + kx2 + 7x + 5 is divided by x + 6 and gives a remainder of −1?

Question

tkhunny · Answer

How's your synthetic division?

mathmale · Answer

Please express exponentiation properly.  It's not x3 + kx2 + 7x + 5, but rather $$x^3+  kx^2+7x+5$$

mathmale · Answer

Next question is what to do with that last term, the 5.  If the remainder is -1, should we add -1 to 5 (to obtain a constant term of 4), or should we subtract -1 from 5 (to obtain a constant term of 6.  Personally I have to experiement to answer my own question.

mathmale · Answer

(x+6) is a factor (given).  Thus, we use -6 as the divisor in synthetic division.  First I assume that 4 as a constant term would result in no remainder:

    -6    /   1            k            7             4
                             -6        -6k+36    36k-258     What value of k would result in a rem.
             ______________________________________       of zero?
                 1          k-6      -6k+43

mathmale · Answer

This test does not necessarily provide the correct value for the unknown, k.  

Would anyone like to propose a more systematic method that would reveal the correct k value more efficiently?

tkhunny · Answer

??  36k - 254 = -1

Solve.

freckles · Answer

The remainder theorem can be used.

freckles · Answer

Let P(x) be a polynomial. 

$$\frac{P(x)}{x-c}=Q(x)+\frac{R}{x-c} \ P(x)=Q(x) (x-c)+R$$
Where R is a constant.

$$P(c)=Q(x)(c-c)+R \ P(c)=Q(x)(0)+R \ P(c)=0+R \ P(c)=R$$

tkhunny · Answer

Yup.  Just did.  Less formally, with the synthetic division, of course.

freckles · Answer

I understand that. Someone said something about finding another way too though so that is why I presented this way. Learning this theorem also means less work. 
Since all you have to do here is solve
P(-6)=-1 for k 
where P is that third degree polynomial given