Shelia is dividing x4 – 4x3 + 4x2 – 6x + 5 by x – 1 using a division table.


Quotient

Divisor
x3
– 3x2
+ x
– 5
x
x4
–3x3
x2
A
– 1
–x3
3x2
B
C
What are the missing values in the table?
A = –5x; B = 1x; C = 5
A = –5x; B = –1x; C = 5
A = 5x; B = –1x; C = 5
A = 5x; B = 1x; C = 5

1 answer

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.

  1. 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.

  2. Multiply \(x^3\) by the divisor \(x - 1\): \[ x^3(x - 1) = x^4 - x^3 \] Subtract this from the dividend: \[ (x^4 - 4x^3 + 4x^2 - 6x + 5) - (x^4 - x^3) = -3x^3 + 4x^2 - 6x + 5 \]

  3. 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.

  4. Multiply \(-3x^2\) by the divisor: \[ -3x^2(x - 1) = -3x^3 + 3x^2 \] Subtract: \[ (-3x^3 + 4x^2 - 6x + 5) - (-3x^3 + 3x^2) = x^2 - 6x + 5 \]

  5. Third Step: Divide the new leading term \(x^2\) by \(x\): \[ \frac{x^2}{x} = x \] Write \(x\) in the quotient.

  6. Multiply \(x\) by the divisor: \[ x(x - 1) = x^2 - x \] Subtract: \[ (x^2 - 6x + 5) - (x^2 - x) = -5x + 5 \]

  7. Fourth Step: Divide \(-5x\) by \(x\): \[ \frac{-5x}{x} = -5 \] Write \(-5\) in the quotient.

  8. 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.