To determine the values of A, B, and C in the division table while dividing \(x^4 - 4x^3 + 4x^2 - 6x + 5\) by \(x - 1\), we can perform polynomial long division.
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First Step: Divide the leading term of the dividend by the leading term of the divisor. \[ \frac{x^4}{x} = x^3 \] Write \(x^3\) in the quotient.
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Multiply \(x^3\) by the divisor \(x - 1\): \[ x^3(x - 1) = x^4 - x^3 \] Subtract this from the dividend: \[ (x^4 - 4x^3 + 4x^2 - 6x + 5) - (x^4 - x^3) = -3x^3 + 4x^2 - 6x + 5 \]
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Second Step: Repeat the division with the new polynomial \(-3x^3 + 4x^2 - 6x + 5\). \[ \frac{-3x^3}{x} = -3x^2 \] Write \(-3x^2\) in the quotient.
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Multiply \(-3x^2\) by the divisor: \[ -3x^2(x - 1) = -3x^3 + 3x^2 \] Subtract: \[ (-3x^3 + 4x^2 - 6x + 5) - (-3x^3 + 3x^2) = x^2 - 6x + 5 \]
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Third Step: Divide the new leading term \(x^2\) by \(x\): \[ \frac{x^2}{x} = x \] Write \(x\) in the quotient.
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Multiply \(x\) by the divisor: \[ x(x - 1) = x^2 - x \] Subtract: \[ (x^2 - 6x + 5) - (x^2 - x) = -5x + 5 \]
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Fourth Step: Divide \(-5x\) by \(x\): \[ \frac{-5x}{x} = -5 \] Write \(-5\) in the quotient.
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Multiply \(-5\) by the divisor: \[ -5(x - 1) = -5x + 5 \] Subtract: \[ (-5x + 5) - (-5x + 5) = 0 \]
Now we have the complete quotient: \[ \text{Quotient} = x^3 - 3x^2 + x - 5 \]
Identifying values:
- From the division table, we can identify:
- \(A = -5\)
- \(B = 1\) (coefficient of \(x\))
- \(C = 5\) (constant term)
Thus, the correct set of values is:
- A = -5; B = 1; C = 5
The closest answer matches:
- A = -5x; B = 1x; C = 5.