Question

find the quotient and remainder using synthetic division: \frac{x^4 - 6x^3 + 4x + 0}{x - 1}

the quotient is x^4 - 5x^3 - 5x^2 - x

the remainder is -1

Explanation:

🆕 New Concept Discovered: Synthetic Division
A shortcut method to divide polynomials by linear factors.

Step 1: Set up the synthetic division table

To divide \( \frac{x^4 - 6x^3 + 4x + 0}{x - 1} \), we first identify the divisor's root and the coefficients of the dividend.

• The divisor is \( x - 1 \), so the root we use is \( c = 1 \).
• The dividend is \( x^4 - 6x^3 + 0x^2 + 4x + 0 \). We must include a \( 0 \) coefficient for the missing \( x^2 \) term.
• The coefficients are: \( 1, -6, 0, 4, 0 \).

We set up the table:

  1 |  1   -6    0    4    0
    |
    -------------------------

Step 2: Perform the synthetic division steps

1. Bring down the first coefficient, \( 1 \):

  1 |  1   -6    0    4    0
    |
    -------------------------
       1

1. Multiply \( 1 \) (the root) by \( 1 \) (the bottom value) to get \( 1 \). Place this under \( -6 \) and add:

$$ -6 + 1 = -5 $$

  1 |  1   -6    0    4    0
    |       1
    -------------------------
       1   -5

1. Multiply \( 1 \) by \( -5 \) to get \( -5 \). Place this under \( 0 \) and add:

$$ 0 + (-5) = -5 $$

  1 |  1   -6    0    4    0
    |       1   -5
    -------------------------
       1   -5   -5

1. Multiply \( 1 \) by \( -5 \) to get \( -5 \). Place this under \( 4 \) and add:

$$ 4 + (-5) = -1 $$

  1 |  1   -6    0    4    0
    |       1   -5   -5
    -------------------------
       1   -5   -5   -1

1. Multiply \( 1 \) by \( -1 \) to get \( -1 \). Place this under \( 0 \) and add:

$$ 0 + (-1) = -1 $$

  1 |  1   -6    0    4    0
    |       1   -5   -5   -1
    -------------------------
       1   -5   -5   -1 | -1

Step 3: Interpret the results

The bottom row values \( 1, -5, -5, -1 \) represent the coefficients of the quotient polynomial, which will be one degree lower than the original dividend (\( x^4 \)). The final value, \( -1 \), is the remainder.

• The quotient coefficients correspond to:

$$ 1x^3 - 5x^2 - 5x - 1 $$

• The remainder is:

$$ -1 $$

Answer:

• The quotient is \( x^3 - 5x^2 - 5x - 1 \)
• The remainder is \( -1 \)