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Divide. If there is a remainder, include it as a simplified fraction.\newline(u3+5u2+4u)÷(u+1)(u^3 + 5u^2 + 4u) \div (u + 1)\newline______

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Q. Divide. If there is a remainder, include it as a simplified fraction.\newline(u3+5u2+4u)÷(u+1)(u^3 + 5u^2 + 4u) \div (u + 1)\newline______
  1. Divide by uu: We will use polynomial long division to divide (u3+5u2+4u)(u^3 + 5u^2 + 4u) by (u+1)(u + 1). First, we divide the first term of the dividend, u3u^3, by the first term of the divisor, uu, to get the first term of the quotient. u3÷u=u2u^3 \div u = u^2
  2. Subtract and Simplify: We multiply the divisor (u+1)(u + 1) by the first term of the quotient u2u^2 and subtract the result from the dividend.\newline(u+1)(u2)=u3+u2(u + 1)(u^2) = u^3 + u^2\newlineSubtract this from the original polynomial:\newline(u3+5u2+4u)(u3+u2)=4u2+4u(u^3 + 5u^2 + 4u) - (u^3 + u^2) = 4u^2 + 4u
  3. Divide by uu: Next, we divide the first term of the new polynomial, 4u24u^2, by the first term of the divisor, uu, to get the next term of the quotient.\newline4u2÷u=4u4u^2 \div u = 4u
  4. Subtract and Simplify: We multiply the divisor (u+1)(u + 1) by the new term of the quotient (4u)(4u) and subtract the result from the new polynomial.\newline(u+1)(4u)=4u2+4u(u + 1)(4u) = 4u^2 + 4u\newlineSubtract this from the new polynomial:\newline(4u2+4u)(4u2+4u)=0(4u^2 + 4u) - (4u^2 + 4u) = 0
  5. Complete the Division: Since we have no remainder, the division is complete. The quotient is the sum of the terms we found: u2+4uu^2 + 4u.

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