Question

Divide using synthetic division.
StartFraction x Superscript 5 Baseline plus 2 x cubed minus 3 Over x minus 1 EndFraction
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Part 1
StartFraction x Superscript 5 Baseline plus 2 x cubed minus 3 Over x minus 1 EndFraction
equals
  
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​(Simplify your​ answer.)

Answer 1

To divide \( \frac{x^5 + 2x^3 - 3}{x - 1} \) using synthetic division, we will follow these steps:

1. **Set up Synthetic Division:** Since we are dividing by \(x - 1\), the root we will use is \(1\).

2. **Prepare the coefficients:** The polynomial \(x^5 + 2x^3 - 3\) can be expressed with coefficients. Since there are no \(x^4\) and \(x^2\) terms in the polynomial, we represent them with a coefficient of \(0\):
- Coefficients: \(1, 0, 2, 0, 0, -3\)

3. **Synthetic Division Process:**

1. Bring down the leading coefficient (1):
\[
\begin{array}{r|rrrrrr}
1 & 1 & 0 & 2 & 0 & 0 & -3 \\
& & 1 & & & & \\
\hline
& 1 & & & & & \\
\end{array}
\]

2. Multiply \(1\) by \(1\) and add it to the next coefficient:
\[
\begin{array}{r|rrrrrr}
1 & 1 & 0 & 2 & 0 & 0 & -3 \\
& & 1 & & & & \\
\hline
& 1 & 1 & & & & \\
\end{array}
\]

3. Multiply \(1\) by \(1\) and add it to the next coefficient:
\[
\begin{array}{r|rrrrrr}
1 & 1 & 0 & 2 & 0 & 0 & -3 \\
& & 1 & 1 & & & \\
\hline
& 1 & 1 & 3 & & & \\
\end{array}
\]

4. Multiply \(1\) by \(3\) and add it to the next coefficient:
\[
\begin{array}{r|rrrrrr}
1 & 1 & 0 & 2 & 0 & 0 & -3 \\
& & 1 & 1 & 3 & & \\
\hline
& 1 & 1 & 3 & 3 & & \\
\end{array}
\]

5. Multiply \(1\) by \(3\) and add it to the next coefficient:
\[
\begin{array}{r|rrrrrr}
1 & 1 & 0 & 2 & 0 & 0 & -3 \\
& & 1 & 1 & 3 & 3 & \\
\hline
& 1 & 1 & 3 & 3 & 3 & \\
\end{array}
\]

6. Multiply \(1\) by \(3\) and add it to the next coefficient:
\[
\begin{array}{r|rrrrrr}
1 & 1 & 0 & 2 & 0 & 0 & -3 \\
& & 1 & 1 & 3 & 3 & 3 \\
\hline
& 1 & 1 & 3 & 3 & 3 & 0 \\
\end{array}
\]

4. **Result:** The bottom row gives us the coefficients of the quotient polynomial and the remainder. The coefficients \(1, 1, 3, 3, 3\) correspond to the polynomial:

\[
x^4 + x^3 + 3x^2 + 3x + 3
\]

5. Since the last number is \(0\), there is no remainder.

Thus, the result of the division:

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