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P(x)=x^(4)-2x^(3)+kx-4
where 
k is an unknown integer.

P(x) divided by 
(x-1) has a remainder of 0 .
What is the value of 
k ?

k=

P(x)=x42x3+kx4 P(x)=x^{4}-2 x^{3}+k x-4 \newlinewhere k k is an unknown integer.\newlineP(x) P(x) divided by (x1) (x-1) has a remainder of 00 .\newlineWhat is the value of k k ?\newlinek= k=

Full solution

Q. P(x)=x42x3+kx4 P(x)=x^{4}-2 x^{3}+k x-4 \newlinewhere k k is an unknown integer.\newlineP(x) P(x) divided by (x1) (x-1) has a remainder of 00 .\newlineWhat is the value of k k ?\newlinek= k=
  1. Apply Remainder Theorem: Apply the Remainder Theorem.\newlineThe Remainder Theorem states that if a polynomial P(x)P(x) is divided by (xc)(x - c) and the remainder is 00, then cc is a root of the polynomial. This means P(c)=0P(c) = 0.
  2. Substitute x=1x = 1: Substitute x=1x = 1 into P(x)P(x).\newlineSince the remainder is 00 when P(x)P(x) is divided by (x1)(x - 1), we substitute x=1x = 1 into the polynomial P(x)P(x) to find the value of kk.\newlineP(1)=(1)42(1)3+k(1)4P(1) = (1)^4 - 2(1)^3 + k(1) - 4
  3. Simplify the expression: Simplify the expression.\newlineP(1)=12+k4P(1) = 1 - 2 + k - 4\newlineP(1)=k5P(1) = k - 5
  4. Set P(1)P(1) equal to 00: Set P(1)P(1) equal to 00. Since P(1)P(1) must be 00 for the remainder to be 00, we set the simplified expression equal to 00. k5=0k - 5 = 0
  5. Solve for k: Solve for k.\newlineAdd 55 to both sides of the equation to solve for k.\newlinek=5k = 5

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