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Answer :
Sure! Let's use synthetic division to divide the polynomial [tex]\(3x^3 + 7x^2 + 0x - 1\)[/tex] by [tex]\(x + 2\)[/tex]. Here's how you can do it step-by-step:
1. Identify the divisor and its root: Since we are dividing by [tex]\(x + 2\)[/tex], we need the root of this divisor, which is [tex]\(-2\)[/tex].
2. Set up the synthetic division table:
- Write down the coefficients of the polynomial: [tex]\(3, 7, 0, -1\)[/tex].
- We will perform synthetic division using the root [tex]\(-2\)[/tex].
3. Perform synthetic division:
- Step 1: Bring down the leading coefficient, 3, to the bottom row.
[tex]\[
\begin{array}{c|cccc}
-2 & 3 & 7 & 0 & -1 \\
& & & & \\
\hline
& 3 & & & \\
\end{array}
\][/tex]
- Step 2: Multiply the number you just brought down (3) by [tex]\(-2\)[/tex] and write the result under the next coefficient (7).
[tex]\[
\begin{array}{c|cccc}
-2 & 3 & 7 & 0 & -1 \\
& & -6 & & \\
\hline
& 3 & & & \\
\end{array}
\][/tex]
- Step 3: Add the result to the next coefficient: [tex]\(7 + (-6) = 1\)[/tex].
[tex]\[
\begin{array}{c|cccc}
-2 & 3 & 7 & 0 & -1 \\
& & -6 & -2 & \\
\hline
& 3 & 1 & & \\
\end{array}
\][/tex]
- Step 4: Repeat the process: Multiply 1 by [tex]\(-2\)[/tex] and write the result, [tex]\(-2\)[/tex], under the next coefficient (0).
[tex]\[
\begin{array}{c|cccc}
-2 & 3 & 7 & 0 & -1 \\
& & -6 & -2 & \\
\hline
& 3 & 1 & -2 & \\
\end{array}
\][/tex]
- Step 5: Add [tex]\(-2\)[/tex] to [tex]\(0\)[/tex]: [tex]\(0 + (-2) = -2\)[/tex].
- Step 6: Multiply [tex]\(-2\)[/tex] by [tex]\(-2\)[/tex] and write the result, 4, under the final coefficient ([tex]\(-1\)[/tex]).
[tex]\[
\begin{array}{c|cccc}
-2 & 3 & 7 & 0 & -1 \\
& & -6 & -2 & 4 \\
\hline
& 3 & 1 & -2 & 3 \\
\end{array}
\][/tex]
- Step 7: Add 4 to [tex]\(-1\)[/tex]: [tex]\(-1 + 4 = 3\)[/tex].
4. Interpret the results:
- The numbers at the bottom row, [tex]\(3, 1, \)[/tex] and [tex]\(-2\)[/tex], are the coefficients of the quotient polynomial.
- The last number, 3, is the remainder.
So, the quotient of dividing [tex]\(3x^3 + 7x^2 - 1\)[/tex] by [tex]\(x + 2\)[/tex] is [tex]\(3x^2 + x - 2\)[/tex] with a remainder of 3.
1. Identify the divisor and its root: Since we are dividing by [tex]\(x + 2\)[/tex], we need the root of this divisor, which is [tex]\(-2\)[/tex].
2. Set up the synthetic division table:
- Write down the coefficients of the polynomial: [tex]\(3, 7, 0, -1\)[/tex].
- We will perform synthetic division using the root [tex]\(-2\)[/tex].
3. Perform synthetic division:
- Step 1: Bring down the leading coefficient, 3, to the bottom row.
[tex]\[
\begin{array}{c|cccc}
-2 & 3 & 7 & 0 & -1 \\
& & & & \\
\hline
& 3 & & & \\
\end{array}
\][/tex]
- Step 2: Multiply the number you just brought down (3) by [tex]\(-2\)[/tex] and write the result under the next coefficient (7).
[tex]\[
\begin{array}{c|cccc}
-2 & 3 & 7 & 0 & -1 \\
& & -6 & & \\
\hline
& 3 & & & \\
\end{array}
\][/tex]
- Step 3: Add the result to the next coefficient: [tex]\(7 + (-6) = 1\)[/tex].
[tex]\[
\begin{array}{c|cccc}
-2 & 3 & 7 & 0 & -1 \\
& & -6 & -2 & \\
\hline
& 3 & 1 & & \\
\end{array}
\][/tex]
- Step 4: Repeat the process: Multiply 1 by [tex]\(-2\)[/tex] and write the result, [tex]\(-2\)[/tex], under the next coefficient (0).
[tex]\[
\begin{array}{c|cccc}
-2 & 3 & 7 & 0 & -1 \\
& & -6 & -2 & \\
\hline
& 3 & 1 & -2 & \\
\end{array}
\][/tex]
- Step 5: Add [tex]\(-2\)[/tex] to [tex]\(0\)[/tex]: [tex]\(0 + (-2) = -2\)[/tex].
- Step 6: Multiply [tex]\(-2\)[/tex] by [tex]\(-2\)[/tex] and write the result, 4, under the final coefficient ([tex]\(-1\)[/tex]).
[tex]\[
\begin{array}{c|cccc}
-2 & 3 & 7 & 0 & -1 \\
& & -6 & -2 & 4 \\
\hline
& 3 & 1 & -2 & 3 \\
\end{array}
\][/tex]
- Step 7: Add 4 to [tex]\(-1\)[/tex]: [tex]\(-1 + 4 = 3\)[/tex].
4. Interpret the results:
- The numbers at the bottom row, [tex]\(3, 1, \)[/tex] and [tex]\(-2\)[/tex], are the coefficients of the quotient polynomial.
- The last number, 3, is the remainder.
So, the quotient of dividing [tex]\(3x^3 + 7x^2 - 1\)[/tex] by [tex]\(x + 2\)[/tex] is [tex]\(3x^2 + x - 2\)[/tex] with a remainder of 3.
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