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Divide. If there is a remainder, include it as a simplified fraction.\newline(4n3+22n212n)÷(n+6)(4n^3 + 22n^2 - 12n) \div (n + 6)\newline______

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Q. Divide. If there is a remainder, include it as a simplified fraction.\newline(4n3+22n212n)÷(n+6)(4n^3 + 22n^2 - 12n) \div (n + 6)\newline______
  1. Divide by First Term: We will use polynomial long division to divide (4n3+22n212n)(4n^3 + 22n^2 - 12n) by (n+6)(n + 6). First, we divide the first term of the dividend, 4n34n^3, by the first term of the divisor, nn, to get the first term of the quotient. 4n3÷n=4n24n^3 \div n = 4n^2
  2. Subtract and Simplify: We multiply the divisor (n+6)(n + 6) by the first term of the quotient (4n2)(4n^2) and subtract the result from the dividend.\newline(4n3+22n212n)(4n2×(n+6))=(4n3+22n212n)(4n3+24n2)(4n^3 + 22n^2 - 12n) - (4n^2 \times (n + 6)) = (4n^3 + 22n^2 - 12n) - (4n^3 + 24n^2)\newlineThis simplifies to 2n212n-2n^2 - 12n.
  3. Divide New Leading Term: Next, we divide the new leading term of the remaining polynomial, 2n2-2n^2, by the first term of the divisor, nn. \newline2n2÷n=2n-2n^2 \div n = -2n
  4. Subtract and Simplify: We multiply the divisor (n+6)(n + 6) by the new term of the quotient (2n)(-2n) and subtract the result from the remaining polynomial.\newline(2n212n)(2n×(n+6))=(2n212n)(2n212n)((-2n^2 - 12n) - (-2n \times (n + 6)) = (-2n^2 - 12n) - (-2n^2 - 12n)(\newlineThis simplifies to \$0\), meaning there is no remainder.
  5. Final Quotient: The quotient we have found is \(4n^2 - 2n\), and there is no remainder.\(\newline\)Therefore, the division is complete.

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