Question
Divide using synthetic division. Write your answer in standard form.
(
5
x
4
+
12
x
3
−
21
x
2
−
40
x
−
12
)
÷
(
x
+
3
)
(5x
4
+12x
3
−21x
2
−40x−12) ÷ (x+3)
Answers
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Answered by
GPT-4o mini
To perform synthetic division of the polynomial \( 5x^4 + 12x^3 - 21x^2 - 40x - 12 \) by \( x + 3 \), we will use the root of \( x + 3 \), which is \( -3 \).
1. **Set up the synthetic division**:
- Write the coefficients of the polynomial: \( 5, 12, -21, -40, -12 \).
- Use \( -3 \) as the divisor.
2. **Synthetic division steps**:
- Write down the coefficients:
\[
\begin{array}{r|rrrrr}
-3 & 5 & 12 & -21 & -40 & -12 \\
& & & & & \\
\end{array}
\]
- Bring down the leading coefficient (5):
\[
\begin{array}{r|rrrrr}
-3 & 5 & 12 & -21 & -40 & -12 \\
& & & & & \\
& 5 & & & & \\
\end{array}
\]
- Multiply \( 5 \) by \( -3 \) and write the result under the next coefficient:
\[
\begin{array}{r|rrrrr}
-3 & 5 & 12 & -21 & -40 & -12 \\
& & -15 & & & \\
& 5 & & & & \\
\end{array}
\]
- Add the second column:
\[
\begin{array}{r|rrrrr}
-3 & 5 & 12 & -21 & -40 & -12 \\
& & -15 & & & \\
& 5 & -3 & & & \\
\end{array}
\]
- Continue this process:
\[
-3 \times -3 = 9 \quad \text{(add to -21:)} \quad -21 + 9 = -12
\]
- Continue this:
\[
\begin{array}{r|rrrrr}
-3 & 5 & 12 & -21 & -40 & -12 \\
& & -15 & 9 & & \\
& 5 & -3 & -12 & & \\
\end{array}
\]
- Multiply \( -12 \) by \( -3 \):
\[
-12 \times -3 = 36
\]
- Add to -40:
\[
-40 + 36 = -4
\]
\[
\begin{array}{r|rrrrr}
-3 & 5 & 12 & -21 & -40 & -12 \\
& & -15 & 9 & 36 & \\
& 5 & -3 & -12 & -4 & \\
\end{array}
\]
- Finally, multiply \( -4 \) by \( -3 \):
\[
-4 \times -3 = 12
\]
- Add to -12:
\[
-12 + 12 = 0
\]
3. **Complete synthetic division**:
The synthetic division is complete:
\[
\begin{array}{r|rrrrr}
-3 & 5 & 12 & -21 & -40 & -12 \\
& & -15 & 9 & 36 & 12 \\
& 5 & -3 & -12 & -4 & 0 \\
\end{array}
\]
- The result of the synthetic division is \( 5x^3 - 3x^2 - 12x - 4 \) (with a remainder of 0).
4. **Write the answer** in standard form:
Thus, the solution to the division of \( 5x^4 + 12x^3 - 21x^2 - 40x - 12 \) by \( x + 3 \) is:
\[
5x^3 - 3x^2 - 12x - 4
\]
1. **Set up the synthetic division**:
- Write the coefficients of the polynomial: \( 5, 12, -21, -40, -12 \).
- Use \( -3 \) as the divisor.
2. **Synthetic division steps**:
- Write down the coefficients:
\[
\begin{array}{r|rrrrr}
-3 & 5 & 12 & -21 & -40 & -12 \\
& & & & & \\
\end{array}
\]
- Bring down the leading coefficient (5):
\[
\begin{array}{r|rrrrr}
-3 & 5 & 12 & -21 & -40 & -12 \\
& & & & & \\
& 5 & & & & \\
\end{array}
\]
- Multiply \( 5 \) by \( -3 \) and write the result under the next coefficient:
\[
\begin{array}{r|rrrrr}
-3 & 5 & 12 & -21 & -40 & -12 \\
& & -15 & & & \\
& 5 & & & & \\
\end{array}
\]
- Add the second column:
\[
\begin{array}{r|rrrrr}
-3 & 5 & 12 & -21 & -40 & -12 \\
& & -15 & & & \\
& 5 & -3 & & & \\
\end{array}
\]
- Continue this process:
\[
-3 \times -3 = 9 \quad \text{(add to -21:)} \quad -21 + 9 = -12
\]
- Continue this:
\[
\begin{array}{r|rrrrr}
-3 & 5 & 12 & -21 & -40 & -12 \\
& & -15 & 9 & & \\
& 5 & -3 & -12 & & \\
\end{array}
\]
- Multiply \( -12 \) by \( -3 \):
\[
-12 \times -3 = 36
\]
- Add to -40:
\[
-40 + 36 = -4
\]
\[
\begin{array}{r|rrrrr}
-3 & 5 & 12 & -21 & -40 & -12 \\
& & -15 & 9 & 36 & \\
& 5 & -3 & -12 & -4 & \\
\end{array}
\]
- Finally, multiply \( -4 \) by \( -3 \):
\[
-4 \times -3 = 12
\]
- Add to -12:
\[
-12 + 12 = 0
\]
3. **Complete synthetic division**:
The synthetic division is complete:
\[
\begin{array}{r|rrrrr}
-3 & 5 & 12 & -21 & -40 & -12 \\
& & -15 & 9 & 36 & 12 \\
& 5 & -3 & -12 & -4 & 0 \\
\end{array}
\]
- The result of the synthetic division is \( 5x^3 - 3x^2 - 12x - 4 \) (with a remainder of 0).
4. **Write the answer** in standard form:
Thus, the solution to the division of \( 5x^4 + 12x^3 - 21x^2 - 40x - 12 \) by \( x + 3 \) is:
\[
5x^3 - 3x^2 - 12x - 4
\]
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