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Divide. If there is a remainder, include it as a simplified fraction.\newline(k3+4k25k)÷(k1)(k^3 + 4k^2 - 5k) \div (k - 1)\newline______

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Q. Divide. If there is a remainder, include it as a simplified fraction.\newline(k3+4k25k)÷(k1)(k^3 + 4k^2 - 5k) \div (k - 1)\newline______
  1. Divide by kk: We will use polynomial long division to divide (k3+4k25k)(k^3 + 4k^2 - 5k) by (k1)(k - 1). First, we divide the first term of the dividend, k3k^3, by the first term of the divisor, kk, to get the first term of the quotient. k3÷k=k2k^3 \div k = k^2
  2. Subtract and Multiply: We multiply the divisor (k1)(k - 1) by the first term of the quotient k2k^2 and subtract the result from the dividend.\newline(k1)×k2=k3k2(k - 1) \times k^2 = k^3 - k^2\newlineSubtract this from the dividend:\newline(k3+4k25k)(k3k2)=4k2k25k=3k25k(k^3 + 4k^2 - 5k) - (k^3 - k^2) = 4k^2 - k^2 - 5k = 3k^2 - 5k
  3. Divide 3k23k^2 by kk: Next, we divide the first term of the new dividend, 3k23k^2, by the first term of the divisor, kk, to get the next term of the quotient.\newline3k2÷k=3k3k^2 \div k = 3k
  4. Subtract and Multiply: We multiply the divisor (k1)(k - 1) by the new term of the quotient (3k)(3k) and subtract the result from the new dividend.\newline(k1)×3k=3k23k(k - 1) \times 3k = 3k^2 - 3k\newlineSubtract this from the new dividend:\newline(3k25k)(3k23k)=5k+3k=2k(3k^2 - 5k) - (3k^2 - 3k) = -5k + 3k = -2k
  5. Divide 2k-2k by kk: Now, we divide the first term of the new dividend, 2k-2k, by the first term of the divisor, kk, to get the next term of the quotient.\newline2k÷k=2-2k \div k = -2
  6. Subtract and Multiply: We multiply the divisor (k1)(k - 1) by the new term of the quotient (2)(-2) and subtract the result from the new dividend.\newline(k1)×(2)=2k+2(k - 1) \times (-2) = -2k + 2\newlineSubtract this from the new dividend:\newline(2k)(2k+2)=2k+2k2=2(-2k) - (-2k + 2) = -2k + 2k - 2 = -2
  7. Final Remainder: Since the degree of the remainder (2-2) is less than the degree of the divisor (k1k - 1), we cannot continue the division process. The remainder is 2-2.

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