To solve the division of
[tex]\[ \frac{3x^3 + 16x^2 + 13x - 7}{x + 4} \][/tex]
using synthetic division, follow these steps:
1. Identify the coefficients of the polynomial:
The polynomial [tex]\(3x^3 + 16x^2 + 13x - 7\)[/tex] has coefficients:
[tex]\[ 3, 16, 13, -7 \][/tex]
2. Determine the root of the divisor:
The divisor is [tex]\(x + 4\)[/tex], which can be written as [tex]\(x - (-4)\)[/tex]. Hence, the root is:
[tex]\[ x = -4 \][/tex]
3. Set up the synthetic division table:
- Place the coefficients in the first row: [tex]\[ 3, 16, 13, -7 \][/tex]
- Write the root [tex]\(-4\)[/tex] on the left side.
4. Begin the synthetic division process:
- Bring down the first coefficient (3) unchanged.
[tex]\[
\begin{array}{r|rrrr}
-4 & 3 & 16 & 13 & -7 \\
& & & & \\
\hline
& 3 & & & \\
\end{array}
\][/tex]
- Multiply this coefficient by the root [tex]\(-4\)[/tex] and write the result under the second coefficient (16):
[tex]\[
\begin{array}{r|rrrr}
-4 & 3 & 16 & 13 & -7 \\
& & -12 & & \\
\hline
& 3 & & & \\
\end{array}
\][/tex]
- Add this result to the second coefficient:
[tex]\[
\begin{array}{r|rrrr}
-4 & 3 & 16 & 13 & -7 \\
& & -12 & & \\
\hline
& 3 & 4 & & \\
\end{array}
\][/tex]
- Repeat the same process: Multiply the new value (4) by [tex]\(-4\)[/tex] and add the result to the next coefficient (13):
[tex]\[
\begin{array}{r|rrrr}
-4 & 3 & 16 & 13 & -7 \\
& & -12 & -16 & \\
\hline
& 3 & 4 & -3 & \\
\end{array}
\][/tex]
- Lastly, multiply this new result (-3) by [tex]\(-4\)[/tex] and add the result to the last coefficient (-7):
[tex]\[
\begin{array}{r|rrrr}
-4 & 3 & 16 & 13 & -7 \\
& & -12 & -16 & 12 \\
\hline
& 3 & 4 & -3 & 5 \\
\end{array}
\][/tex]
5. Interpret the result:
- The numbers at the bottom (except the final number) are the coefficients of the quotient.
- The final number is the remainder.
Therefore, the quotient of the division is:
[tex]\[
3x^2 + 4x - 3
\][/tex]
and the remainder is:
[tex]\[
5
\][/tex]
So, the result of dividing [tex]\(\frac{3x^3 + 16x^2 + 13x - 7}{x + 4}\)[/tex] using synthetic division is:
[tex]\[
\boxed{3x^2 + 4x - 3 \text{ (quotient)}, \ 5 \text{ (remainder)}}
\][/tex]
