To perform the division of the polynomial \(24x^4 + 18x^3 - 9x^2 + 3\) by \(4x^2 - 3x + 1\), we'll use polynomial long division.
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Set up the division: We write \(24x^4 + 18x^3 - 9x^2 + 0x + 3\) under the long division symbol and \(4x^2 - 3x + 1\) on the outside.
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Divide the leading terms:
- Divide the leading term of the dividend \(24x^4\) by the leading term of the divisor \(4x^2\): \[ \frac{24x^4}{4x^2} = 6x^2 \]
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Multiply and subtract:
- Multiply \(6x^2\) by the entire divisor \(4x^2 - 3x + 1\): \[ 6x^2(4x^2 - 3x + 1) = 24x^4 - 18x^3 + 6x^2 \]
- Subtract this result from the original polynomial: \[ (24x^4 + 18x^3 - 9x^2 + 3) - (24x^4 - 18x^3 + 6x^2) = 36x^3 - 15x^2 + 3 \]
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Repeat the process:
- Now we take the new polynomial \(36x^3 - 15x^2 + 3\).
- Divide the leading term \(36x^3\) by the leading term \(4x^2\): \[ \frac{36x^3}{4x^2} = 9x \]
- Multiply \(9x\) by the divisor \(4x^2 - 3x + 1\): \[ 9x(4x^2 - 3x + 1) = 36x^3 - 27x^2 + 9x \]
- Subtract this from \(36x^3 - 15x^2 + 3\): \[ (36x^3 - 15x^2 + 3) - (36x^3 - 27x^2 + 9x) = 12x^2 - 9x + 3 \]
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Repeat again:
- Take \(12x^2 - 9x + 3\).
- Divide \(12x^2\) by \(4x^2\): \[ \frac{12x^2}{4x^2} = 3 \]
- Multiply \(3\) by \(4x^2 - 3x + 1\): \[ 3(4x^2 - 3x + 1) = 12x^2 - 9x + 3 \]
- Subtract: \[ (12x^2 - 9x + 3) - (12x^2 - 9x + 3) = 0 \]
No remainder is left.
- Conclusion: The quotient is: \[ 6x^2 + 9x + 3 \]
Therefore, the result of dividing \(24x^4 + 18x^3 - 9x^2 + 3\) by \(4x^2 - 3x + 1\) is: \[ 6x^2 + 9x + 3 \]