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.

Step-by-Step :

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

  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 ]

  3. 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 ]

  4. 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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