How do I find k if x^4-kx^3-2x^2+x+4 is divided by x-3 and the remainder is 16

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Let  f(x) = x^4-kx^3-2x^2+x+4

If f(x) is divided by x-3 , then the remainder is 16.

Then we could write:

f(x) = (x-3) * R(x) + 16

Now let us substitute with x =3:

==> f(3) = 0 * R(3) + 16 = 16

==> f(3) = 16 .......(1)

Now...

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Let  f(x) = x^4-kx^3-2x^2+x+4

If f(x) is divided by x-3 , then the remainder is 16.

Then we could write:

f(x) = (x-3) * R(x) + 16

Now let us substitute with x =3:

==> f(3) = 0 * R(3) + 16 = 16

==> f(3) = 16 .......(1)

Now let us substitute x=3 in f(x):

f(3) = 3^4 - k* 3^3 -2*3^2 + 3 + 4 = 16

==> 81 - 27k - 18 + 7 = 16

==> -27k = -54

==> k = 2

==> f(x) = x^4 -2x^3 -2x^2 + x + 4

==> f(x) = (x-3)( x^3 + x^2 + x + 4) + 16

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