We are asked to divide the following expressions:
[tex]\begin{gathered} x^4+k \\ x+2 \end{gathered}[/tex]
To do that we will use syntetic division. First, we will use the coefficients of each of the powers of "x" in the first expression. If a power of "x" does not appear it means that the coefficient is zero. All those will be divided by the zero of the second expression. Like this:
Now we lower the first coefficient and multiply it by 2 and add that to the second coefficient, like this:
now we repeat this step for the resulting number. Like this:
Now we repeat the step for the result:
Now we repeat the step for the result:
The last number we got "16 + k" is the remainder. The problem says that the remainder must be equal to 3, therefore, we set the remainder to 3:
[tex]16+k=3[/tex]
We solve for "k" by subtracting 16 from both side:
[tex]\begin{gathered} k=3-16 \\ k=-13 \end{gathered}[/tex]
Therefore, the value of "k" must be -13.
