Question
How to do long division
18x^2-12x+6/3x
18x^2-12x+6/3x
Answers
GPT-4o mini
To perform long division with the polynomial \( 18x^2 - 12x + 6 \) divided by \( 3x \), follow these steps:
1. **Set Up the Long Division**: Write \( 18x^2 - 12x + 6 \) under the long division symbol and \( 3x \) outside.
\[
\begin{array}{r|l}
3x & 18x^2 - 12x + 6 \\
\end{array}
\]
2. **Divide the Leading Terms**: Divide the leading term \( 18x^2 \) by \( 3x \).
\[
\frac{18x^2}{3x} = 6x
\]
3. **Multiply**: Multiply \( 6x \) by \( 3x \) and write this product under the original polynomial.
\[
6x \cdot 3x = 18x^2
\]
4. **Subtract**: Subtract this product from the original polynomial.
\[
18x^2 - 12x + 6 - 18x^2 = -12x + 6
\]
5. **Bring Down the Next Term**: (In this case, there's no next term to bring down, as the remainder is already \( -12x + 6 \).)
6. **Divide Again**: Now divide the leading term of the new polynomial (\( -12x \)) by \( 3x \).
\[
\frac{-12x}{3x} = -4
\]
7. **Multiply Again**: Multiply \( -4 \) by \( 3x \) and write this under the polynomial.
\[
-4 \cdot 3x = -12x
\]
8. **Subtract Again**: Subtract this from the current polynomial \( -12x + 6 \).
\[
-12x + 6 - (-12x) = 6
\]
Now, we are left with a remainder of \( 6 \).
9. **Final Result**: The result of the long division is:
\[
\text{Quotient: } 6x - 4 \quad \text{Remainder: } 6
\]
So, you can express the result of the division as:
\[
\frac{18x^2 - 12x + 6}{3x} = 6x - 4 + \frac{6}{3x} = 6x - 4 + \frac{2}{x}
\]
This means that \( 18x^2 - 12x + 6 \) divided by \( 3x \) gives us a quotient of \( 6x - 4 \) with a remainder of \( 6 \) or an expression of \( 6x - 4 + \frac{2}{x} \).
1. **Set Up the Long Division**: Write \( 18x^2 - 12x + 6 \) under the long division symbol and \( 3x \) outside.
\[
\begin{array}{r|l}
3x & 18x^2 - 12x + 6 \\
\end{array}
\]
2. **Divide the Leading Terms**: Divide the leading term \( 18x^2 \) by \( 3x \).
\[
\frac{18x^2}{3x} = 6x
\]
3. **Multiply**: Multiply \( 6x \) by \( 3x \) and write this product under the original polynomial.
\[
6x \cdot 3x = 18x^2
\]
4. **Subtract**: Subtract this product from the original polynomial.
\[
18x^2 - 12x + 6 - 18x^2 = -12x + 6
\]
5. **Bring Down the Next Term**: (In this case, there's no next term to bring down, as the remainder is already \( -12x + 6 \).)
6. **Divide Again**: Now divide the leading term of the new polynomial (\( -12x \)) by \( 3x \).
\[
\frac{-12x}{3x} = -4
\]
7. **Multiply Again**: Multiply \( -4 \) by \( 3x \) and write this under the polynomial.
\[
-4 \cdot 3x = -12x
\]
8. **Subtract Again**: Subtract this from the current polynomial \( -12x + 6 \).
\[
-12x + 6 - (-12x) = 6
\]
Now, we are left with a remainder of \( 6 \).
9. **Final Result**: The result of the long division is:
\[
\text{Quotient: } 6x - 4 \quad \text{Remainder: } 6
\]
So, you can express the result of the division as:
\[
\frac{18x^2 - 12x + 6}{3x} = 6x - 4 + \frac{6}{3x} = 6x - 4 + \frac{2}{x}
\]
This means that \( 18x^2 - 12x + 6 \) divided by \( 3x \) gives us a quotient of \( 6x - 4 \) with a remainder of \( 6 \) or an expression of \( 6x - 4 + \frac{2}{x} \).