The value of k so that x^44x^3+5x^22x+k is divisible by x^22x+2 is: A) 0 B) 2 C) 1 D) 2
Question
The value of $k$ so that $x^{4}4x^{3}+5x^{2}2x+k$ is divisible by $x^{2}2x+2$ is:
A) 0 B) 2 C) 1 D) 2
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Answer
To determine the value of $k$ such that the polynomial ( x^{4}  4x^{3} + 5x^{2}  2x + k ) is divisible by ( x^{2}  2x + 2 ), we need to divide the polynomial by ( x^{2}  2x + 2 ) and ensure the remainder is zero.
StepbyStep :

Write Down the Polynomial and Divisor: Given polynomial: ( x^{4}  4x^{3} + 5x^{2}  2x + k ) Divisor: ( x^{2}  2x + 2 )

Perform Polynomial Division:

First Division Step: [ \frac{x^{4}}{x^{2}} = x^{2} ] Multiply ( x^{2} ) by the divisor: [ x^{2} \cdot (x^{2}  2x + 2) = x^{4}  2x^{3} + 2x^{2} ] Subtract from the original polynomial: [ (x^{4}  4x^{3} + 5x^{2}  2x + k)  (x^{4}  2x^{3} + 2x^{2}) = 2x^{3} + 3x^{2}  2x + k ]

Second Division Step: [ \frac{2x^{3}}{x^{2}} = 2x ] Multiply ( 2x ) by the divisor: [ 2x \cdot (x^{2}  2x + 2) = 2x^{3} + 4x^{2}  4x ] Subtract from the resulting polynomial: [ (2x^{3} + 3x^{2}  2x + k)  (2x^{3} + 4x^{2}  4x) = x^{2} + 2x + k ]

Third Division Step: [ \frac{x^{2}}{x^{2}} = 1 ] Multiply ( 1 ) by the divisor: [ 1 \cdot (x^{2}  2x + 2) = x^{2} + 2x  2 ] Subtract from the resulting polynomial: [ (x^{2} + 2x + k)  (x^{2} + 2x  2) = k + 2 ]


Ensure the Polynomial is Divisible: The polynomial ( x^{4}  4x^{3} + 5x^{2}  2x + k ) will be divisible by ( x^{2}  2x + 2 ) if and only if the remainder is zero: [ k + 2 = 0 ]

Solve for ( k ): [ k = 2 ]
Conclusion:
The value of ( k ) is ( \boxed{2} ).
Thus, the correct answer is Option B: 2.
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