Question

performing synthetic division with remainders

find the quotient.
\\((2x^4 - 3x^3 - 6x^2 + 11x + 8) \div (x - 2)\\)

• \\(2x^3 - 7x^2 + 8x - 5 + \frac{18}{x-2}\\)
• \\(2x^3 + x^2 - 4x + 3 + \frac{14}{x-2}\\)
• \\(4x^3 + 2x^2 - 8x + 6 + \frac{14}{x-2}\\)
• \\(4x^3 - 14x^2 + 16x - 10 + \frac{18}{x-2}\\)

find the quotient.
\\((5x^4 - 3x^2 + 4) \div (x + 1)\\)

• \\(5x^3 - 5x^2 + 2x - 2 + \frac{6}{x+1}\\)
• \\(5x^3 - 8x^2 + 8 - \frac{4}{x+1}\\)
• \\(5x^3 - 5x + 2 + \frac{2}{x+1}\\)
• \\(5x^2 + 2x + \frac{2}{x+1}\\)

Explanation:

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

Step 1: Set up the synthetic division

To divide \( (5x^4 - 3x^2 + 4) \div (x + 1) \), we first write down the coefficients of the dividend. We must include a \( 0 \) coefficient for any missing terms (the \( x^3 \) and \( x^1 \) terms):

$$ 5x^4 + 0x^3 - 3x^2 + 0x + 1 $$

The coefficients are:

$$ 5, \quad 0, \quad -3, \quad 0, \quad 4 $$

Since we are dividing by \( x + 1 \), the value we use for division is \( c = -1 \).

Step 2: Perform the division steps

Bring down the first coefficient, then multiply by \( -1 \) and add repeatedly:

 -1 |   5    0   -3    0    4
    |       -5    5   -2    2
    -------------------------
        5   -5    2   -2    6

• Bring down \( 5 \).
• Multiply: \( 5 \times (-1) = -5 \). Add: \( 0 + (-5) = -5 \).
• Multiply: \( -5 \times (-1) = 5 \). Add: \( -3 + 5 = 2 \).
• Multiply: \( 2 \times (-1) = -2 \). Add: \( 0 + (-2) = -2 \).
• Multiply: \( -2 \times (-1) = 2 \). Add: \( 4 + 2 = 6 \).

Step 3: Write the final quotient

The numbers at the bottom represent the coefficients of the quotient polynomial (which is one degree lower than the original polynomial) and the final number is the remainder:

• Coefficients: \( 5, -5, 2, -2 \)
• Remainder: \( 6 \)

This translates to:

$$ 5x^3 - 5x^2 + 2x - 2 + \frac{6}{x+1} $$

Answer:

The correct option is the first one:

$$ 5x^3 - 5x^2 + 2x - 2 + \frac{6}{x+1} $$