SOLUTION: Determine the value of m so that when f(x) is divided by (x-4) the remainder is -8. f(x)=3x^2+mx+4

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Question 1092373: Determine the value of m so that when f(x) is divided by (x-4) the remainder is -8. f(x)=3x^2+mx+4
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
There are two ways to do this problem.

Method 1:

The remainder theorem states: 

When a polynomial f(x) is divided by (x-r), the remainder is f(r).

Therefore f(4) = -8

 3(4)² + m(4) + 4 = -8
   3(16) + 4m + 4 = -8
      48 + 4m + 4 = -8
          52 + 4m = -8
               4m = -60
                m = -15

Method 2:

You can also do it by synthetic division

4 | 3       m         4
  |        12     4(m+12)
    3     m+12   4+4(m+12)

Then set the remainder equal to -8

         4+4(m+12) = -8
           4+4m+48 = -8
             52+4m = -8
                4m = -60
                 m = -15

Edwin
Answer by ikleyn(52834)   (Show Source): You can put this solution on YOUR website!
.
According to the Remainder theorem, 


    if  f(x) gives the remainder -8  when is divided by (x-4),  then the value f(4) is equal to -8:  f(4) = -8.



So, from the condition, you have THIS  equation to find m:

    3*4^2 +m*4 + 4 = -8.


Simplify and solve for m:

    3*16 + 4m + 4 = -8  ====>  4m = -3*16 - 4 - 8  ====>  4m = -60  ====>  m =  = -15.


Answer.  m = -15.

On the Remainder theorem see the lessons
    - Divisibility of polynomial f(x) by binomial (x-a) and the Remainder theorem
    - Solved problems on the Remainder thoerem
in this site.


The first lesson contains the Remainder theorem (its formulation and the proof): 

    Theorem   (the remainder theorem)

    1. The remainder of division the polynomial    by the binomial    is equal to the value    of the polynomial. 

    2. The binomial    divides the polynomial    if and only if the value of    is the root of the polynomial  ,  i.e.  .

    3. The binomial    factors the polynomial    if and only if the value of    is the root of the polynomial  ,  i.e.  .


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