To perform long division with the polynomials \( 4x^3 + 11x^2 + 30x + 184 \) divided by \( 4x + 34x + 3 \), let's first clarify the polynomials involved.
The polynomial to be divided is:
\[ 4x^3 + 11x^2 + 30x + 184 \]
And the divisor is:
\[ 4x + 34x + 3 = 38x + 3 \]
Steps of Long Division:
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Divide the first term of the dividend by the first term of the divisor: \[ \frac{4x^3}{38x} = \frac{4}{38}x^2 = \frac{2}{19}x^2 \]
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Multiply the entire divisor by this result \(\frac{2}{19}x^2\): \[ \left(\frac{2}{19}x^2\right)(38x + 3) = \frac{2}{19} \cdot 38x^3 + \frac{2}{19} \cdot 3x^2 = 4x^3 + \frac{6}{19}x^2 \]
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Subtract this from the original polynomial: \[ \left(4x^3 + 11x^2 + 30x + 184\right) - \left(4x^3 + \frac{6}{19}x^2\right) \] The \(4x^3\) terms cancel out: \[ \left(11x^2 - \frac{6}{19}x^2\right) + 30x + 184 \] To subtract the \(x^2\) terms, we can convert \(11\) to a fraction: \[ 11 = \frac{209}{19} \quad \Rightarrow \quad \frac{209}{19} - \frac{6}{19} = \frac{203}{19} \] So we have: \[ \frac{203}{19}x^2 + 30x + 184 \]
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Next, divide the leading term \(\frac{203}{19}x^2\) by \(38x\): \[ \frac{\frac{203}{19}x^2}{38x} = \frac{203}{19 \cdot 38}x = \frac{203}{722}x \]
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Multiply the divisor by \(\frac{203}{722}x\): \[ \left(\frac{203}{722}x\right)(38x + 3) = \frac{203}{722}(38x^2 + 3x) = \frac{203 \cdot 38}{722}x^2 + \frac{203 \cdot 3}{722}x \]
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Calculate \(203 \cdot 38\): \[ 203 \cdot 38 = 7714 \] Therefore: \[ \frac{7714}{722}x^2 + \frac{609}{722}x \]
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Subtract this from the previously simplified form: \[ \left(\frac{203}{19}x^2 + 30x + 184\right) - \left(\frac{7714}{722}x^2 + \frac{609}{722}x\right) \] To perform this subtraction, it's useful to get a common denominator. The common denominator will be \(722\): \[ \text{Convert }\frac{203}{19}x^2 = \frac{203 \cdot 38}{722}x^2 = \frac{7714}{722}x^2 \] \[ \text{Convert }30x = \frac{30 \cdot 722}{722}x = \frac{21660}{722}x \]
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The subtraction now looks like this: \[ \left(\frac{7714}{722} - \frac{7714}{722}\right)x^2 + \left(\frac{21660}{722} - \frac{609}{722}\right)x + 184 \]
After subtraction, the \(x^2\) terms disappear: \[ \frac{21660 - 609}{722}x + 184 = \frac{21051}{722}x + 184 \]
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Finally, divide \(\frac{21051}{722}x + 184\) by \(38x + 3\). Since the degrees of the terms are less than the divisor, we stop here.
The Result:
The final quotient from the polynomial long division is: \[ \frac{2}{19}x^2 + \frac{203}{722}x + \text{Remainder} \]
Where the remainder is: \[ \frac{21051}{722}x + 184 \]
Thus the division: \[ \frac{4x^3 + 11x^2 + 30x + 184}{38x + 3} = \frac{2}{19}x^2 + \frac{203}{722}x + \frac{21051}{722} + \frac{184}{38x + 3} \]
The full result in long division format is: \[ \text{Quotient: } \frac{2}{19}x^2 + \frac{203}{722}x \text{ and Remainder: } \frac{21051}{722}x + 184. \]
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