Question

divide using synthetic division.

\\\frac{x^5 + 2x^4 - x^3 - 3x^2 - 4x - 1}{x - 2}\\

\\\frac{x^5 + 2x^4 - x^3 - 3x^2 - 4x - 1}{x - 2} = \boxed{\quad}\\
(simplify your answer.)

Explanation:

Identify the setup for synthetic division

We divide \(x^5 + 2x^4 - x^3 - 3x^2 - 4x - 1\) by \(x - 2\).
The divisor is \(x - c\), so \(c = 2\).
The coefficients of the dividend are:

$$1, \quad 2, \quad -1, \quad -3, \quad -4, \quad -1$$

Perform synthetic division calculations

We set up the synthetic division table:

$$
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$$

The calculations are:

• Bring down the first coefficient: \(1\)
• Multiply by \(2\): \(1 \times 2 = 2\). Add: \(2 + 2 = 4\)
• Multiply by \(2\): \(4 \times 2 = 8\). Add: \(-1 + 8 = 7\)
• Multiply by \(2\): \(7 \times 2 = 14\). Add: \(-3 + 14 = 11\)
• Multiply by \(2\): \(11 \times 2 = 22\). Add: \(-4 + 22 = 18\)
• Multiply by \(2\): \(18 \times 2 = 36\). Add: \(-1 + 36 = 35\)

Interpret the resulting coefficients

The quotient coefficients are \(1, 4, 7, 11, 18\) and the remainder is \(35\).
Since the dividend is of degree 5, the quotient is of degree 4:

$$x^4 + 4x^3 + 7x^2 + 11x + 18$$

Write the final simplified expression

We express the result in the form \(Q(x) + \frac{R(x)}{D(x)}\):

$$x^4 + 4x^3 + 7x^2 + 11x + 18 + \frac{35}{x-2}$$

Answer:

\(\frac{x^5 + 2x^4 - x^3 - 3x^2 - 4x - 1}{x - 2} =\) <blank>\(x^4 + 4x^3 + 7x^2 + 11x + 18 + \frac{35}{x-2}\)</blank>