To divide the polynomial [tex]\(x^2 + 9\)[/tex] by [tex]\(x + 4\)[/tex] using synthetic division, follow these steps:
1. Identify the coefficients of the dividend (the polynomial you are dividing):
The polynomial [tex]\(x^2 + 9\)[/tex] can be written as:
[tex]\[
x^2 + 0x + 9
\][/tex]
Thus, the coefficients are [tex]\(1, 0,\)[/tex] and [tex]\(9\)[/tex].
2. Identify the root of the divisor:
The divisor is [tex]\(x + 4\)[/tex]. Setting [tex]\(x + 4 = 0\)[/tex] gives [tex]\(x = -4\)[/tex]. So, the root is [tex]\(-4\)[/tex].
3. Set up the synthetic division table:
Write down the root [tex]\(-4\)[/tex] on the left, and the coefficients [tex]\(1, 0,\)[/tex] and [tex]\(9\)[/tex] on the right:
[tex]\[
\begin{array}{r|rrr}
-4 & 1 & 0 & 9 \\
\hline
& & &
\end{array}
\][/tex]
4. Perform the synthetic division:
- Carry down the leading coefficient [tex]\(1\)[/tex] to the bottom row:
[tex]\[
\begin{array}{r|rrr}
-4 & 1 & 0 & 9 \\
\hline
& 1 &
\end{array}
\][/tex]
- Multiply this result by the root [tex]\(-4\)[/tex] and write the result under the next coefficient:
[tex]\(1 \times (-4) = -4\)[/tex]
[tex]\[
\begin{array}{r|rrr}
-4 & 1 & 0 & 9 \\
\hline
& 1 & -4
\end{array}
\][/tex]
- Add the value from the previous step [tex]\(-4\)[/tex] to the next coefficient [tex]\(0\)[/tex]:
[tex]\(0 + (-4) = -4\)[/tex]
[tex]\[
\begin{array}{r|rrr}
-4 & 1 & 0 & 9 \\
\hline
& 1 & -4 &
\end{array}
\][/tex]
- Repeat the process: multiply the last result [tex]\(-4\)[/tex] by the root [tex]\(-4\)[/tex] and write the result under the next coefficient:
[tex]\(-4 \times (-4) = 16\)[/tex]
[tex]\[
\begin{array}{r|rrr}
-4 & 1 & 0 & 9 \\
\hline
& 1 & -4 & 16
\end{array}
\][/tex]
- Add the value from the previous step [tex]\(16\)[/tex] to the next coefficient [tex]\(9\)[/tex]:
[tex]\(9 + 16 = 25\)[/tex]
[tex]\[
\begin{array}{r|rrr}
-4 & 1 & 0 & 9 \\
\hline
& 1 & -4 & 25
\end{array}
\][/tex]
5. Interpret the results:
The numbers on the bottom row, except the last one, represent the coefficients of the quotient. The last number is the remainder.
Thus, the quotient is [tex]\(x - 4\)[/tex] and the remainder is 25.
Therefore, the division of [tex]\(x^2 + 9\)[/tex] by [tex]\(x + 4\)[/tex] using synthetic division yields:
[tex]\[
\frac{x^2 + 9}{x + 4} = x - 4 \quad \text{with a remainder of } 25.
\][/tex]
