To find the quotient of \( 2x^3 + 3x^2 + 5x - 4 \) divided by \( x^2 + x + 1 \), we will perform polynomial long division.
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Division Step: Divide the leading term of the dividend by the leading term of the divisor: \[ \frac{2x^3}{x^2} = 2x \]
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Multiply: Multiply the entire divisor \( x^2 + x + 1 \) by \( 2x \): \[ 2x(x^2 + x + 1) = 2x^3 + 2x^2 + 2x \]
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Subtract: Subtract this result from the original polynomial: \[ (2x^3 + 3x^2 + 5x - 4) - (2x^3 + 2x^2 + 2x) = (3x^2 - 2x^2) + (5x - 2x) - 4 = x^2 + 3x - 4 \]
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Next Division Step: Now we repeat the process with the new polynomial \( x^2 + 3x - 4 \): \[ \frac{x^2}{x^2} = 1 \]
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Multiply: Multiply the divisor by \( 1 \): \[ 1(x^2 + x + 1) = x^2 + x + 1 \]
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Subtract: Subtract this from \( x^2 + 3x - 4 \): \[ (x^2 + 3x - 4) - (x^2 + x + 1) = (3x - x) + (-4 - 1) = 2x - 5 \]
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Final Output: The degree of the remainder \( 2x - 5 \) is less than the degree of the divisor \( x^2 + x + 1 \). Thus, we stop here.
The final result of the long division is: \[ \text{Quotient: } 2x + 1 \quad \text{Remainder: } 2x - 5 \]
In summary: \[ \frac{2x^3 + 3x^2 + 5x - 4}{x^2 + x + 1} = 2x + 1 + \frac{2x - 5}{x^2 + x + 1} \]
So the quotient is \( 2x + 1 \).
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