To divide the polynomial [tex]\( x^4 + 2x^2 - 1 \)[/tex] by [tex]\( x + 1 \)[/tex] using synthetic division, follow these steps:
1. Setup the coefficients:
The polynomial [tex]\( x^4 + 2x^2 - 1 \)[/tex] can be written as [tex]\( x^4 + 0x^3 + 2x^2 + 0x - 1 \)[/tex], with coefficients [tex]\([1, 0, 2, 0, -1]\)[/tex].
2. Use the root of the divisor:
Since we are dividing by [tex]\( x + 1 \)[/tex], the root of the divisor is [tex]\( -1 \)[/tex].
3. Synthetic Division:
- Write down the coefficients: [tex]\([1, 0, 2, 0, -1]\)[/tex].
- Bring down the leading coefficient (1) to the bottom row.
[tex]\[
\begin{array}{r|rrrrr}
-1 & 1 & 0 & 2 & 0 & -1 \\
\hline
& 1 & & & & \\
\end{array}
\][/tex]
- Multiply the root [tex]\(-1\)[/tex] by the value just written down (1) and write the result (-1) under the next coefficient (0).
[tex]\[
\begin{array}{r|rrrrr}
-1 & 1 & 0 & 2 & 0 & -1 \\
\hline
& 1 & -1 & & & \\
\end{array}
\][/tex]
- Add the column: [tex]\( (0) + (-1) = -1 \)[/tex].
[tex]\[
\begin{array}{r|rrrrr}
-1 & 1 & 0 & 2 & 0 & -1 \\
\hline
& 1 & -1 & & & \\
& & & -1 & & \\
\end{array}
\][/tex]
- Multiply the root [tex]\(-1\)[/tex] by the value just written down (-1) and write the result (1) under the next coefficient (2).
[tex]\[
\begin{array}{r|rrrrr}
-1 & 1 & 0 & 2 & 0 & -1 \\
\hline
& 1 & -1 & 3 & & \\
\end{array}
\][/tex]
- Add the column: [tex]\( (2) + (1) = 3 \)[/tex].
[tex]\[
\begin{array}{r|rrrrr}
-1 & 1 & 0 & 2 & 0 & -1 \\
\hline
& 1 & -1 & 3 & -3 & \\
\end{array}
\][/tex]
- Multiply the root [tex]\(-1\)[/tex] by the value just written down (3) and write the result (-3) under the next coefficient (0).
[tex]\[
\begin{array}{r|rrrrr}
-1 & 1 & 0 & 2 & 0 & -1 \\
\hline
& 1 & -1 & 3 & -3 & 2 \\
\end{array}
\][/tex]
- Add the column: [tex]\( (0) + (-3) = -3 \)[/tex].
[tex]\[
\begin{array}{r|rrrrr}
-1 & 1 & 0 & 2 & 0 & -1 \\
\hline
& 1 & -1 & 3 & -3 & 2 \\
\end{array}
\][/tex]
- Finally, multiply the root [tex]\(-1\)[/tex] by the value just written down (-3) and write the result (3) under the last coefficient (-1).
[tex]\[
\begin{array}{r|rrrrr}
-1 & 1 & 0 & 2 & 0 & -1 \\
\hline
& 1 & -1 & 3 & -3 & 2 \\
\end{array}
\][/tex]
- Add the column: [tex]\( (-1) + (3) = 2 \)[/tex].
4. Interpretion:
The bottom row [tex]\([1, -1, 3, -3, 2]\)[/tex] gives us the coefficients of the quotient polynomial and the remainder.
Therefore, the quotient is:
[tex]\[
1x^3 - 1x^2 + 3x - 3
\][/tex]
and the remainder is 2.
So, the quotient and remainder of the division [tex]\( \frac{x^4 + 2x^2 - 1}{x + 1} \)[/tex] are:
[tex]\[ x^3 - x^2 + 3x - 3 + \frac{2}{x+1} \][/tex]
