To perform synthetic division for the polynomial [tex]\(2x^3 + 6x^2 - 5x + 7\)[/tex] by the divisor [tex]\(x + 3\)[/tex], we follow these steps:
1. Identify the coefficients of the polynomial:
The polynomial [tex]\(2x^3 + 6x^2 - 5x + 7\)[/tex] has coefficients:
[tex]\[
[2, 6, -5, 7]
\][/tex]
2. Identify [tex]\(a\)[/tex] where the divisor is [tex]\(x - (-3)\)[/tex]:
Here, [tex]\(a = -3\)[/tex].
3. Set up the synthetic division:
Write the coefficients of the polynomial in a row. We'll use [tex]\(a = -3\)[/tex] for the synthetic division process.
4. Synthetic Division Process:
[tex]\[
\begin{array}{r|rrrr}
-3 & 2 & 6 & -5 & 7 \\
\hline
& & \textcolor{red}{2} & \textcolor{red}{0} & \textcolor{red}{-5} & \textcolor{red}{22} \\
& & 2 \\
& -6 & \\
& \textcolor{red}{6} & \\
& -0 & \\
& \textcolor{red}{0} & \\
& -0 & \\
& \textcolor{red}{-5} & \\
& 15 & \\
& \textcolor{red}{22} & \\
& -66 & \\
& \textcolor{red}{/ -3}
\end{array}
\][/tex]
5. Perform the Division:
- Bring down the leading coefficient: [tex]\(2\)[/tex].
- Multiply [tex]\(2\)[/tex] by [tex]\(a = -3\)[/tex] and write the result under the next coefficient.
- Add the result to the next coefficient: [tex]\(6 + (-6) = 0\)[/tex].
- Multiply [tex]\(0\)[/tex] by [tex]\(-3\)[/tex] and write the result under the next coefficient.
- Add the result to the next coefficient: [tex]\(-5 + 0 = -5\)[/tex].
- Multiply [tex]\(-5\)[/tex] by [tex]\(-3\)[/tex] and write the result under the next coefficient.
- Add the result to the next coefficient: [tex]\(7 + 15 = 22\)[/tex].
6. Identify the Quotient and Remainder:
After completing the procedure:
- The coefficients of the quotient polynomial are [tex]\(2, 0, -5\)[/tex], which means the quotient is [tex]\(2x^2 + 0x - 5\)[/tex].
- The remainder after division is [tex]\(22\)[/tex].
So, the division of [tex]\(2x^3 + 6x^2 - 5x + 7\)[/tex] by [tex]\(x + 3\)[/tex] gives a quotient of [tex]\(2x^2 - 5\)[/tex] with a remainder of [tex]\(22\)[/tex].
7. Write the Result in the Required Form:
Combine the quotient and the remainder to express the result in the form
[tex]\[
2x^2 - 5 + \frac{22}{x + 3}
\][/tex]
