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)   (Show Source): You can put this solution on YOUR website!



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


k=-4
Answer by ikleyn(52915)   (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    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.  .


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.



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