Question

use synthetic division to find the remainder.\\(\frac{x^{4}}{x + 1}\\)

Explanation:

Step1: Identify the root of the divisor

The divisor is \(x + 1\), so we set \(x+1 = 0\), which gives \(x=-1\). This is the value we will use in synthetic division. The dividend is \(x^{4}\), which can be written as \(x^{4}+0x^{3}+0x^{2}+0x + 0\) to include all the coefficients of the powers of \(x\) from \(x^{4}\) down to the constant term.

Step2: Set up synthetic division

Write the root \(-1\) on the left, and the coefficients of the dividend \(1, 0, 0, 0, 0\) (for \(x^{4},x^{3},x^{2},x^{1},x^{0}\) respectively) in a row:

\[

$$\begin{array}{r|rrrrr} -1 & 1 & 0 & 0 & 0 & 0 \\ \ & \ & \ & \ & \ & \ \\ \hline \ & \ & \ & \ & \ & \ \\ \end{array}$$

\]

Step3: Perform synthetic division

Bring down the first coefficient (1) as it is:

\[

$$\begin{array}{r|rrrrr} -1 & 1 & 0 & 0 & 0 & 0 \\ \ & \ & -1 & 1 & -1 & 1 \\ \hline \ & 1 & -1 & 1 & -1 & 1 \\ \end{array}$$

\]

• Multiply the brought - down number (1) by the root (-1): \(1\times(-1)=-1\). Add this to the next coefficient (0): \(0+(-1)=-1\).
• Multiply the new number (-1) by the root (-1): \((-1)\times(-1) = 1\). Add this to the next coefficient (0): \(0 + 1=1\).
• Multiply the new number (1) by the root (-1): \(1\times(-1)=-1\). Add this to the next coefficient (0): \(0+(-1)=-1\).
• Multiply the new number (-1) by the root (-1): \((-1)\times(-1)=1\). Add this to the last coefficient (0): \(0 + 1=1\).

The last number in the bottom row is the remainder. According to the Remainder Theorem, when we divide a polynomial \(f(x)\) by \(x - a\), the remainder is \(f(a)\). Here, \(f(x)=x^{4}\) and \(a=-1\), so \(f(-1)=(-1)^{4}=1\), which matches the result from synthetic division.

Answer:

The remainder is \(\boldsymbol{1}\)