To solve the polynomial equation [tex]\( n^3 + n^2 + 4n + 4 = 0 \)[/tex] using synthetic division, we start by testing possible roots. One common choice is testing [tex]\( n = -1 \)[/tex].
### Step-by-Step Solution:
1. Setup the coefficients and the test root:
The polynomial is [tex]\( n^3 + n^2 + 4n + 4 \)[/tex]. The coefficients are [1, 1, 4, 4], and the test root is [tex]\( n = -1 \)[/tex].
2. Perform synthetic division:
We will perform the synthetic division by following these steps:
- Write down the coefficients: 1, 1, 4, 4.
- Write the test root [tex]\( n = -1 \)[/tex] to the left.
- Bring down the first coefficient (1) directly below the line.
Here's the setup:
[tex]\[
\begin{array}{r|rrrr}
-1 & 1 & 1 & 4 & 4 \\
\hline
& 1 & & & \\
\end{array}
\][/tex]
3. Complete the synthetic division process:
- Multiply the first coefficient by the test root and write the result under the second coefficient: [tex]\(1 \cdot (-1) = -1\)[/tex].
- Add this result to the second coefficient: [tex]\(1 + (-1) = 0\)[/tex].
Update the table:
[tex]\[
\begin{array}{r|rrrr}
-1 & 1 & 1 & 4 & 4 \\
\hline
& 1 & 0 & & \\
& & -1 & & \\
\end{array}
\][/tex]
- Continue with the next set of operations:
- Multiply 0 by -1: [tex]\(0 \cdot -1 = 0\)[/tex].
- Add this to the third coefficient: [tex]\(4 + 0 = 4\)[/tex].
Update the table:
[tex]\[
\begin{array}{r|rrrr}
-1 & 1 & 1 & 4 & 4 \\
\hline
& 1 & 0 & 4 & \\
& & -1 & 0 & \\
\end{array}
\][/tex]
- Continue with the last step:
- Multiply 4 by -1: [tex]\(4 \cdot -1 = -4\)[/tex].
- Add this to the fourth coefficient: [tex]\(4 + (-4) = 0\)[/tex].
Final table:
[tex]\[
\begin{array}{r|rrrr}
-1 & 1 & 1 & 4 & 4 \\
\hline
& 1 & 0 & 4 & 0 \\
& & -1 & 0 & -4 \\
\end{array}
\][/tex]
The last row (below the line) shows the coefficients: 1, 0, 4, 0.
4. Interpretation of the result:
The result of the synthetic division is [tex]\([1, 0, 4, 0]\)[/tex] with a remainder of 0. This indicates that [tex]\(n = -1\)[/tex] is indeed a root of the polynomial.
5. Result:
The synthetic division confirms that [tex]\( n = -1 \)[/tex] is a root of the polynomial equation [tex]\( n^3 + n^2 + 4n + 4 = 0 \)[/tex]. The quotient from this synthetic division is [tex]\( n^2 + 4 \)[/tex], implying the factorization:
[tex]\[
(n + 1)(n^2 + 4) = 0
\][/tex]
Solving [tex]\( n^2 + 4 = 0 \)[/tex] gives [tex]\( n = \pm 2i \)[/tex].
### Final Solution:
The roots of the polynomial [tex]\( n^3 + n^2 + 4n + 4 = 0 \)[/tex] are:
[tex]\[
n = -1, \quad n = 2i, \quad n = -2i.
\][/tex]
