Remainder Theorem to Solve a Polynomial

1) Determine the value of k so that when P(x) = x(cubed) + kx(squared) - 2x(squared) + 1 is divided by x + 2, the remainder is 5.

2) Using long division, determine the remainder when P(x) is divided by x-3

The polynomial will not be, in general, in the form you gave. I am taking the general form

P(x) = x^4 + kx^2 - 2x^2 + 1 leaves remainder 5 when divided by x+2

Refer to Remainder theorem which states that the remainder when x-a divides a polynomial P(x) is equal to P(a)

So, we should have P(-2) = 5 -> (-2)^4 + k(-2)^2 - 2(-2)^2 + 1 = 5
That gives 16 +4k - 8 + 1 = 5
This gives k = -1

Now, the polynomial is P(x) = x^4 - x^2 - 2x^2 + 1 = x^4 -3x^2 + 1
Using the long division,

x - 3)x^4 - 3x^2 + 1(x^3 + 3x^2 + 6x + 17
x^4 - 3x^3
_______________
3x^3 - 3x^2
3x^3 - 9x^2
________________
6x^2 + 1
6x^2 - 18x
___________
18x + 1
18x - 54
________
55

Hence the remainder is 55
This can be verified P(3) = (3)^4 - 3(3)^2 + 1 = 81 - 27 + 1 = 55

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