When `x^4 - x^3 - kx^2 - 13x + 2`

is divided by x+3, the remainder is the same as when it is divided by x-5. Determine the value of k.

x^4-x^3-kx^2-13x+2

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+kx2*****

`(x^4 - x^3- kx^2 - 13x + 2)/ (x+3)` = R

`(x^4 - x^3 - kx^2 - 13x + 2)/(x+5)` =R

This means that when x=-3 (from x-3)AND when x=5 (from x+5) we have the same remainder. Substitute ,thus:

`(-3)^4 -(-3)^3-k(-3)^2-13(-3)+2=(5)^4-(5)^3-k(5)^2-13(5)+2`

They equal each other because R=R

81-(-27)-9k -(-39)+2=625 -125-25k-65+2

Care with the negative symbols:

81+27-9k+39+2= 437-25k

`therefore` 149 - 9k= 437 -25k

`therefore` -9k+25k = 437-149

`therefore` 16k=288

`therefore` **k=18**

**Sources:**

The answer if the equation is changed to `+kx^2` changes to a negative answer. Thus:

81 +27 +9k +39+2 = 625 -125 +25k -65 + 2

`therefore` **k= - 18**

`f(x)=x^4-x^3-k x^2-13x+2`

calculate f(-3) and f(5)

put f(-3)=f(5)

i.e.

81+27-9k+39+2=625-125-25k-65+2

147-9k=435-25k

25k-9k=435-147

16k=288

k=18

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