To perform long division with the polynomial \( 18x^2 - 12x + 6 \) divided by \( 3x \), follow these steps:
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Set Up the Long Division: Write \( 18x^2 - 12x + 6 \) under the long division symbol and \( 3x \) outside.
\[ \begin{array}{r|l} 3x & 18x^2 - 12x + 6 \ \end{array} \]
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Divide the Leading Terms: Divide the leading term \( 18x^2 \) by \( 3x \).
\[ \frac{18x^2}{3x} = 6x \]
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Multiply: Multiply \( 6x \) by \( 3x \) and write this product under the original polynomial.
\[ 6x \cdot 3x = 18x^2 \]
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Subtract: Subtract this product from the original polynomial.
\[ 18x^2 - 12x + 6 - 18x^2 = -12x + 6 \]
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Bring Down the Next Term: (In this case, there's no next term to bring down, as the remainder is already \( -12x + 6 \).)
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Divide Again: Now divide the leading term of the new polynomial (\( -12x \)) by \( 3x \).
\[ \frac{-12x}{3x} = -4 \]
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Multiply Again: Multiply \( -4 \) by \( 3x \) and write this under the polynomial.
\[ -4 \cdot 3x = -12x \]
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Subtract Again: Subtract this from the current polynomial \( -12x + 6 \).
\[ -12x + 6 - (-12x) = 6 \]
Now, we are left with a remainder of \( 6 \).
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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 \( 6x - 4 + \frac{2}{x} \).
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