Question
use the long division method to find the result when 3x^3+17x^2+30x+18 is divided by x+3
dont get another wrong answer
dont get another wrong answer
Answers
GPT-4o mini
To perform the long division of \( 3x^3 + 17x^2 + 30x + 18 \) by \( x + 3 \), we can follow these steps:
1. **Setup the Division**: Write the dividend \( 3x^3 + 17x^2 + 30x + 18 \) under the long division symbol, and the divisor \( x + 3 \) outside.
2. **Divide the Leading Terms**: Divide the leading term of the dividend \( 3x^3 \) by the leading term of the divisor \( x \):
\[
\frac{3x^3}{x} = 3x^2
\]
Write \( 3x^2 \) above the division line.
3. **Multiply and Subtract**: Multiply \( 3x^2 \) by the entire divisor \( x + 3 \):
\[
3x^2 \cdot (x + 3) = 3x^3 + 9x^2
\]
Now subtract this result from the original polynomial:
\[
(3x^3 + 17x^2 + 30x + 18) - (3x^3 + 9x^2) = (17x^2 - 9x^2) + 30x + 18 = 8x^2 + 30x + 18
\]
4. **Repeat the Process**: Now, repeat the process with the new polynomial \( 8x^2 + 30x + 18 \).
- Divide \( 8x^2 \) by \( x \):
\[
\frac{8x^2}{x} = 8x
\]
Write \( 8x \) above the division line next to \( 3x^2 \).
- Multiply \( 8x \) by \( x + 3 \):
\[
8x \cdot (x + 3) = 8x^2 + 24x
\]
Subtract this from \( 8x^2 + 30x + 18 \):
\[
(8x^2 + 30x + 18) - (8x^2 + 24x) = (30x - 24x) + 18 = 6x + 18
\]
5. **Continue**: Now repeat with \( 6x + 18 \).
- Divide \( 6x \) by \( x \):
\[
\frac{6x}{x} = 6
\]
Write \( 6 \) above the line next to \( 3x^2 + 8x \).
- Multiply \( 6 \) by \( x + 3 \):
\[
6 \cdot (x + 3) = 6x + 18
\]
Subtract:
\[
(6x + 18) - (6x + 18) = 0
\]
At this point, there is no remainder. The final result of the division is:
\[
3x^2 + 8x + 6
\]
Therefore, when \( 3x^3 + 17x^2 + 30x + 18 \) is divided by \( x + 3 \), the quotient is \( 3x^2 + 8x + 6 \) with a remainder of \( 0 \).
So the final answer is:
\[
\text{Quotient: } 3x^2 + 8x + 6
\]
1. **Setup the Division**: Write the dividend \( 3x^3 + 17x^2 + 30x + 18 \) under the long division symbol, and the divisor \( x + 3 \) outside.
2. **Divide the Leading Terms**: Divide the leading term of the dividend \( 3x^3 \) by the leading term of the divisor \( x \):
\[
\frac{3x^3}{x} = 3x^2
\]
Write \( 3x^2 \) above the division line.
3. **Multiply and Subtract**: Multiply \( 3x^2 \) by the entire divisor \( x + 3 \):
\[
3x^2 \cdot (x + 3) = 3x^3 + 9x^2
\]
Now subtract this result from the original polynomial:
\[
(3x^3 + 17x^2 + 30x + 18) - (3x^3 + 9x^2) = (17x^2 - 9x^2) + 30x + 18 = 8x^2 + 30x + 18
\]
4. **Repeat the Process**: Now, repeat the process with the new polynomial \( 8x^2 + 30x + 18 \).
- Divide \( 8x^2 \) by \( x \):
\[
\frac{8x^2}{x} = 8x
\]
Write \( 8x \) above the division line next to \( 3x^2 \).
- Multiply \( 8x \) by \( x + 3 \):
\[
8x \cdot (x + 3) = 8x^2 + 24x
\]
Subtract this from \( 8x^2 + 30x + 18 \):
\[
(8x^2 + 30x + 18) - (8x^2 + 24x) = (30x - 24x) + 18 = 6x + 18
\]
5. **Continue**: Now repeat with \( 6x + 18 \).
- Divide \( 6x \) by \( x \):
\[
\frac{6x}{x} = 6
\]
Write \( 6 \) above the line next to \( 3x^2 + 8x \).
- Multiply \( 6 \) by \( x + 3 \):
\[
6 \cdot (x + 3) = 6x + 18
\]
Subtract:
\[
(6x + 18) - (6x + 18) = 0
\]
At this point, there is no remainder. The final result of the division is:
\[
3x^2 + 8x + 6
\]
Therefore, when \( 3x^3 + 17x^2 + 30x + 18 \) is divided by \( x + 3 \), the quotient is \( 3x^2 + 8x + 6 \) with a remainder of \( 0 \).
So the final answer is:
\[
\text{Quotient: } 3x^2 + 8x + 6
\]