SOLUTION: use synthetic division or other methods to find the value of k that makes the linear expression a factor of the given expression. x^2-kx+4;x+2

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Question 1157080: use synthetic division or other methods to find the value of k that makes the linear expression a factor of the given expression.
x^2-kx+4;x+2
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
k=-4


Doing through synthetic division:
x+2 as possible factor, root to check -2 
-2   |    1    -k       4
     |
     |         -2      2k+4
     |______________________________
          1   -k-2    2k+8

2k%2B8=0%7D%2C+requirement+if+-2+is+a+root.%0D%0A%7B%7B%7B2k=-8
k=-4

Answer by ikleyn(52909) About Me  (Show Source):
You can put this solution on YOUR website!
.

The other method is to apply the Remainder theorem.


The remainder theorem says that the binomial (x+2) divide a polynomial if and only id the (-2) is the root of the polynomial.


So, you replace x in the given polynomial by -2 and write

    (-2)^2 - k*(-2) + 4 = 0, 

or, which is the same

    4  + 2k + 4 = 0,

    2k = -8,

     k = -8/2 = -4, 


and it is your ANSWER

Solved.

--------------------

   Theorem   (the remainder theorem)
   1. The remainder of division the polynomial  f%28x%29  by the binomial  x-a  is equal to the value  f%28a%29  of the polynomial.
   2. The binomial  x-a  divides the polynomial  f%28x%29  if and only if the value of  a  is the root of the polynomial  f%28x%29,  i.e.  f%28a%29+=+0.
   3. The binomial  x-a  factors the polynomial  f%28x%29  if and only if the value of  a  is the root of the polynomial  f%28x%29,  i.e.  f%28a%29+=+0.


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.


Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic
"Divisibility of polynomial f(x) by binomial (x-a). The Remainder theorem".

Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.