To find [tex]\((2x^4 - 3x^3 - 20x - 21) \div (x - 3)\)[/tex] using synthetic division, follow these steps:
1. Identify the coefficients of the polynomial.
The polynomial [tex]\(2x^4 - 3x^3 + 0x^2 - 20x - 21\)[/tex] has coefficients [tex]\(2, -3, 0, -20, -21\)[/tex].
2. Identify the root of the divisor [tex]\(x - 3\)[/tex].
The root is [tex]\(3\)[/tex].
3. Set up the synthetic division process:
- Write the coefficients: [tex]\(2, -3, 0, -20, -21\)[/tex].
- Write the root [tex]\(3\)[/tex] to the left.
4. Perform the synthetic division step by step:
- Bring down the leading coefficient [tex]\(2\)[/tex].
[tex]\[
\begin{array}{c|ccccc}
3 & 2 & -3 & 0 & -20 & -21 \\
\hline
& 2 & & & & \\
\end{array}
\][/tex]
- Multiply the root [tex]\(3\)[/tex] by the number just written below the line [tex]\(2\)[/tex] and write the result under the next coefficient.
[tex]\[
\begin{array}{c|ccccc}
3 & 2 & -3 & 0 & -20 & -21 \\
\hline
& 2 & 6 & & & \\
\end{array}
\][/tex]
- Add the column: [tex]\(-3 + 6 = 3\)[/tex].
[tex]\[
\begin{array}{c|ccccc}
3 & 2 & -3 & 0 & -20 & -21 \\
\hline
& 2 & 6 & 3 & & \\
\end{array}
\][/tex]
- Repeat the process:
Multiply [tex]\(3\)[/tex] by the number just written below the line [tex]\(3\)[/tex] and write the result under the next coefficient.
[tex]\[
\begin{array}{c|ccccc}
3 & 2 & -3 & 0 & -20 & -21 \\
\hline
& 2 & 6 & 3 & 9 & \\
\end{array}
\][/tex]
- Add the column: [tex]\(0 + 9 = 9\)[/tex].
[tex]\[
\begin{array}{c|ccccc}
3 & 2 & -3 & 0 & -20 & -21 \\
\hline
& 2 & 6 & 3 & 9 & \\
\end{array}
\][/tex]
- Repeat the process:
Multiply [tex]\(3\)[/tex] by the number just written below the line [tex]\(9\)[/tex] and write the result under the next coefficient.
[tex]\[
\begin{array}{c|ccccc}
3 & 2 & -3 & 0 & -20 & -21 \\
\hline
& 2 & 6 & 3 & 9 & 27 \\
\end{array}
\][/tex]
- Add the column: [tex]\(-20 + 27 = 7\)[/tex].
[tex]\[
\begin{array}{c|ccccc}
3 & 2 & -3 & 0 & -20 & -21 \\
\hline
& 2 & 6 & 3 & 9 & 7 \\
\end{array}
\][/tex]
- Multiply [tex]\(3\)[/tex] by the number just written below the line [tex]\(7\)[/tex] and write the result under the next coefficient.
[tex]\[
\begin{array}{c|ccccc}
3 & 2 & -3 & 0 & -20 & -21 \\
\hline
& 2 & 6 & 3 & 9 & 7 & 21 \\
\end{array}
\][/tex]
- Add the column: [tex]\(-21 + 21 = 0\)[/tex].
[tex]\[
\begin{array}{c|ccccc}
3 & 2 & -3 & 0 & -20 & -21 \\
\hline
& 2 & 6 & 3 & 9 & 7 & 0 \\
\end{array}
\][/tex]
5. Interpret the results:
- The values below the line (except the last one) are the coefficients of the quotient.
- The last value [tex]\(0\)[/tex] is the remainder.
The quotient is [tex]\(2x^3 + 3x^2 + 9x + 7\)[/tex] and the remainder is [tex]\(0\)[/tex].
So, the correct answer is [tex]\(\boxed{2x^3 + 3x^2 + 9x + 7}\)[/tex].
Thus, the correct answer is:
A. [tex]\(2 x^3+3 x^2+9 x+7\)[/tex].
